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Talks of D1 Dynamics

Viviane Baladi, CNRS & Sorbonne Université, France

THE FRACTIONAL SUSCEPTIBILITY FUNCTION FOR THE QUADRATIC FAMILY

Joint work with Daniel Smania

Date: 2019-09-18 (Wednesday); Time: 10:40-11:20; Location: building A-3/A-4, room 103.

Abstract

For \(t\) in a set \(\Omega\) of positive measure, maps in the quadratic family \( f_t(x)=t-x^2 \) admit an SRB measure \(\mu_t\). On the one hand, the dependence of \(\mu_t\) on \(t\) has been shown [1] to be no better than \(1/2\) Hölder, on a subset of \(\Omega\), for \(t_0\) a suitable Misiurewicz-Thurston parameter. On the other hand, the susceptibility function \(\Psi_t(z)\), whose value at \(z=1\) is a candidate for the derivative of \(\mu_t\) with respect to \(t\), has been shown [2] to admit a holomorphic extension at \(z=1\) for \(t=t_0\). Our goal is to resolve this paradox. For this, we introduce and study a fractional susceptibility function.

References

  1. V. Baladi, M. Benedicks, and D. Schnellmann, Whitney Hölder continuity of the SRB measure for transversal families of smooth unimodal maps, Invent. Math. 201 (2015), 773-844.
  2. Y. Jiang, D. Ruelle, Analyticity of the susceptibility function for unimodal Markovian maps of the interval, Nonlinearity 18 (2005), 2447-2453.

Tien-Cuong Dinh, National University of Singapore, Singapore

UNIQUE ERGODICITY FOR FOLIATIONS ON COMPACT KAEHLER SURFACES

Joint work with Viet-Anh Nguyen and Nessim Sibony

Date: 2019-09-20 (Friday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

Let \(F\) be a holomorphic foliation by Riemann surfaces on a compact Kaehler surface. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed \((1,1)\)-current, or equivalently, no invariant measure. Then there exists a unique (up to a multiplicative constant) positive ddc-closed \((1,1)\)-current directed by \(F\), or equivalently, a unique harmonic measure. This is a very strong ergodic property showing that all leaves of \(F\) have the same asymptotic behavior. Our proof uses an extension of the theory of densities to a new class of currents. A complete description of the cone of directed positive ddc-closed \((1,1)\)-currents (i.e. harmonic measures) is also given when \(F\) admits directed positive closed currents (i.e. invariant measures).

Manfred Einsiedler, ETH Zürich, Switzerland

MEASURE RIGIDITY FOR HIGHER RANK DIAGONALIZABLE ACTIONS

Joint work with Elon Lindenstrauss

Date: 2019-09-19 (Thursday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

We review old and recent measure rigidity results for higher rank diagonalizable actions on homogeneous spaces and contrast these results with the rank one and unipotent case. After this we consider higher rank actions on irreducible arithmetic quotients of \(\operatorname{SL}_2(\mathbb{R})^k\) for \(k\geq 2\). If the quotient is compact, positive entropy of an ergodic invariant measure \(\mu\) implies algebraicity of \(\mu\) with semisimple stabiliser. For non-compact quotients there are more possibilities. The main novelty here is that the acting group does not have to be maximal or in a special position. The main new idea is to use a quantitative recurrence phenomenon to transport positivity of entropy for one acting element to another.

Michael Hochman, Hebrew University of Jerusalem, Israel

EQUIDISTRIBUTION FOR COMMUTING MAPS

Date: 2019-09-17 (Tuesday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

In two classical papers circa 1960, J. Cassels and W. Schmidt proved that a.e. numbers in the ternary Cantor set (with respect to Cantor-Lebesgue measure) eqidistributes for Lebesgue measure under the map \(Tx=bx \bmod 1\), whenever \(b\) is an integer that is not a power of \(3\). This phenomenon has since been established in much greater generality on the interval, e.g. Host's theorem, according to which one can replace Cantor-Lebesgue measure by any \(\times 3\)-ergodic measure of positive entropy, provided \(\gcd(3,b)=1\). In this talk I will describe a new and heuristically simple proof of such results, and then discuss how it can be extended to give new results in multi-dimensional settings.

Vadim Kaloshin, University of Maryland, College Park, USA

ON DYNAMICAL SPECTRAL RIGIDITY OF PLANAR DOMAINS

Date: 2019-09-18 (Wednesday); Time: 11:25-12:05; Location: building A-3/A-4, room 103.

Abstract

Consider a convex domain on the plane and  the associated billiard inside. The length spectrum is the closure  of the union of perimeters of all period orbits. The length spectrum is  closely related to the Laplace spectrum, through so-called the wave trace. The well-known question popularized by M. Kac: "Can you hear  the shape of a drum?" asks if the Laplace spectrum determines a domain  up to isometry. We call a domain dynamically spectrally rigid (DSR) if any  smooth deformation preserving the length spectrum is an isometry. During  the talk I will discuss recent results on DSR of convex planar domains.

Raphaël Krikorian, CNRS & Université de Cergy-Pontoise, France

ON THE DIVERGENCE OF BIRKHOFF NORMAL FORMS

Date: 2019-09-16 (Monday); Time: 14:15-14:55; Location: building A-3/A-4, room 103.

Abstract

A real analytic hamiltonian or a real analytic exact symplectic diffeomorphism admitting a non resonant elliptic fixed point is always formally conjugated to a formal integrable system, its Birkhoff Normal Form (BNF). Siegel proved in 1954 that the formal conjugation reducing a hamiltonian to its BNF is in general divergent and Hakan Eliasson has asked whether the BNF itself could be divergent. Perez-Marco proved in 2001 that for any fixed non resonant frequency vector the following dichotomy holds: either any real analytic hamiltonian system admitting this frequency vector at the origin has a convergent BNF or for a prevalent set of hamiltonians admitting this frequency vector the BNF generically diverges. It is possible to exhibit examples of hamiltonian systems with diverging BNF (X. Gong 2012 or the recent examples of B. Fayad in 4 degrees of freedom). The aim of this talk is to give a complete answer to the question of the divergence of the BNF (in the setting of exact symplectic diffeomorphisms): for any non resonant frequency vector, the BNF of a real analytic exact symplectic diffeomorphism admitting this frequency vector at the origin, is in general divergent. This theorem is the consequence of the remarkable fact that the convergence of the formal object that is the BNF has dynamical consequences, in particular an abnormal abundance of invariant tori.

Jens Marklof, University of Bristol, UK

KINETIC THEORY FOR THE LOW-DENSITY LORENTZ GAS

Joint work with Andreas Strombergsson

Date: 2019-09-19 (Thursday); Time: 15:00-15:40; Location: building A-3/A-4, room 103.

Abstract

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers, whose radii are small compared to their mean separation. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that its macroscopic transport properties should be governed by a linear Boltzmann equation. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of this lecture is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strombergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers' centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

Mark Pollicott, University of Warwick, UK

INFLECTION POINTS FOR LYAPUNOV SPECTRA

Joint work with Oliver Jenkinson and Polina Vytnova

Date: 2019-09-16 (Monday); Time: 15:00-15:40; Location: building A-3/A-4, room 103.

Abstract

The Lyapunov spectra for a dynamical system describes the size (Hausdorff dimension) of the set of points which have a given Lyapunov exponent. H. Weiss conjectured that the associated graph is convex, but Iommi and Kiwi constructed a simple counter example. We explore this problem further, constructing examples with any given number of points of inflection.

Grzegorz Świątek, Warsaw University of Technology, Poland

MANDELBROT SET SEEN BY HARMONIC MEASURE: THE SIMILARITY MAP

Joint work with Jacek Graczyk

Date: 2019-09-20 (Friday); Time: 15:00-15:40; Location: building A-3/A-4, room 103.

Abstract

We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci \({\cal M}_d\) for unicritical polynomials \(f_c(z)=z^d+c\). It is known that these parameters are structurally unstable and have stochastic dynamics. In [3] it was shown that for \(c\) from a set of full harmonic measure in \(\partial{\cal M}_d\) there exists a quasi-conformal similarity map \(\Upsilon_{c}\) between phase and parameter spaces which is conformal at \(c\). In a recent work [2] we prove \(C^{1+\frac{\alpha}{d}-\epsilon}\)-conformality, \(\alpha = \text{HD}({\cal J}_{c})\), of \(\Upsilon_{c}(z):\mathbb{C}\mapsto \mathbb{C}\) at typical \(c\in \partial {\cal M}_d\) and establish that globally quasiconformal similarity maps \(\Upsilon_{c}(z)\), \(c\in \partial {\cal M}_d\), are \(C^1\)-conformal along external rays landing at \(c\) in \(\mathbb{C}\setminus {\cal J}_{c}\) mapping onto the corresponding rays of \({\cal M}_d\). This conformal equivalence leads to a proof that the \(z\)-derivative of the similarity map \(\Upsilon_{c}(z)\) at typical \(c\in \partial {\cal M}_d\) is equal to \(1/{\cal T}'(c)\), where \[ {\cal T}(c)=\sum_{n=0}^{\infty}\left( D_z\left[f_{c}^n(z)\right]_{z=c}\right)^{-1}\] is the transversality function previously studied by Benedicks-Carleson and Levin, see [1, 4]. There are additonal geometric consequences of these results. A typical external radius of the connectedness locus is contained in an asymptotically very nearly linear twisted angle, but nevertheless passes through infinitely many increasingly narrow straits.

References

  1. M. Benedicks, L. Carleson, On iterations of \(1-ax^2\) on \((-1,1)\), Ann. of Math. 122 (1985), 1-25.
  2. J. Graczyk, G. Świątek, Analytic structures and harmonic measure at bifurcation locus, arXiv 1904.09434 (2019).
  3. J. Graczyk, G. Świątek, Fine structure of connectedness loci, Math. Ann. 369 (2017), 49-108.
  4. G. Levin, An analytical approach to the Fatou conjecture, Fund. Math. 171 (2002), 177-196.

Jan Boroński, AGH University of Science and Technology, Poland

ALL MINIMAL CANTOR SYSTEMS ARE SLOW

Joint work with Jiří Kupka and Piotr Oprocha

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building A-3/A-4, room 103.

Abstract

A Cantor set is a 0-dimensional compact metric space without isolated points, and Cantor system is a dynamical system on the Cantor set. A minimal system is the one that has all orbits dense. We are interested in the following question: Can every minimal Cantor system be embedded into \(\mathbb{R}\) with vanishing derivative everywhere? A particular instance of that question was raised by Samuel Petite at the Workshop on Aperiodic Patterns in Crystals, Numbers and Symbols that took place in Lorentz Center in June of 2017, who asked if expansive minimal Cantor systems have this property. It was conjectured that the expansive systems lack such a property, because some kind of expanding must take place in these systems. In contrast, I shall discuss a postive answer to the above question on all Cantor minimal systems, obtained in [2]. There are more reasons for which this result seems surprising. By the Margulis-Ruelle inequality the topological entropy of a piecewise Lipschitz differentiable map \(f\), with an invariant measure \(\mu\), is bounded from above by the integral over the support of \(\mu\) of the Lyapunov characteristic of \(f\). In the case of derivative zero, all Lyapunov exponents, and as a result Lyapunov characteristic of \(f\) are all equal to \(0\). Therefore it is natural to expect that vanishing derivative on an invariant set will imply zero entropy on that set. Such an intuition was supported by the zero entropy examples in [3] and [1]. However our result shows that no such connection exists. Note that if \(f\) is \(C^1\) on \(\mathbb{R}\) and \(f(P)\subset P\) for some perfect compact subset \(P\subset \mathbb{R}\), then there is \(x\in P\) with \(|f'(x)|\geq 1\) (see [4] for a nice survey on this and related topics). For systems with positive entropy it is also a consequence of Margulis-Ruelle inequality mentioned above, so in this case the map is not even Lipschitz continuous. This gives rise to the following question: Can the differentiable extensions of minimal Cantor systems to \(\mathbb{R}\), guaranteed by our result, be additionally required to be \(\alpha\)-Hölder continuous for some \(0<\alpha<1\)?

References

  1. J. Boroński, J. Kupka, P. Oprocha, Edrei's Conjecture Revisited, Ann. Henri Poincaré 19 (2018), 267-281.
  2. J. Boroński, J. Kupka, P. Oprocha, All Minimal Cantor Systems Are Slow, arXiv:1902.10641.
  3. K.C. Ciesielski, J. Jasiński, An auto-homeomorphism of a Cantor set with derivative zero everywhere, J. Math. Anal. Appl. 434 (2016), 1267-1280.
  4. K.C. Ciesielski, J. Seoane-Sepúlveda, Differentiability versus continuity: restriction and extension theorems and monstrous examples, Bull. Amer. Math. Soc. (N.S.) 56(2) (2019), 211-260.

Henk Bruin, Universität Wien, Austria

ON UNIMODAL INVERSE LIMIT SPACES

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building A-3/A-4, room 103.

Abstract

Inverse limit spaces of unimodal maps are used to model attractors of maps (on the plane). As such, the classification of unimodal inverse limit spaces has an impact on out understanding of e.g. Henon-like attractors. In this talk I want to give an update on how topological properties of unimodal inverse limit spaces relate to dynamical properties of the underlying map.

Jernej Činč, AGH University of Science and Technology, Poland & University of Ostrava, Czech Republic

PRIME ENDS DYNAMICS IN PARAMETRISED FAMILIES OF ROTATIONAL ATTRACTORS

Joint work with Jan P. Boroński and Xiao-Chuan Liu

Date: 2019-09-19 (Thursday); Time: 17:45-18:05; Location: building A-3/A-4, room 103.

Abstract

The prime ends rotation number induced by surface homeomorphisms restricted to open domains is one of the important tools in the study of boundary dynamics. Parametrised families of dynamical systems can provide a clearer view of both the surface dynamics and the boundary dynamics in many situations. Our study [1] serves as a contribution in this direction, by providing new examples in various contexts, by investigating the prime ends rotation numbers arising from parametrized BBM embeddings of inverse limits of topological graphs [2].

First, motivated by a topological version of the Poincaré-Bendixson Theorem obtained recently by Koropecki and Passeggi [4], we show the existence of homeomorphisms of \(S^2\) with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux.

Second, with the help of a reduced Arnold's family we construct a parametrised family of Birkhoff-like cofrontier attractors, where except for countably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This contrasts with the negative resolution of Walker's Conjecture from [5] by Koropecki, Le Calvez and Nassiri [3], and implies that our examples induce Denjoy homeomorphisms on the circles of prime ends.

Third, answering a question of Boyland, we show that there exists a non-transitive Birkhoff-like attracting cofrontier which is obtained from a BBM embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor so that the two induced prime ends rotation numbers are exactly the two endpoints of the rotation interval.

References

  1. J. Boroński, J. Činč, X.-C. Liu, Prime ends dynamics in parametrised families of rotational attractors, arXiv:1906.04640 [math.DS] (2019).
  2. P. Boyland, A. de Carvalho, T. Hall, Prime ends dynamics in parametrised families of rotational attractors, Bull. Lond. Math. Soc. 45(5) (2013), 1075–1085.
  3. A. Koropecki, P. Le Calvez, M. Nassiri, Prime ends rotation numbers and periodic points, Duke Math. J. 164 (2015), 403-472.
  4. A. Koropecki, A. Passeggi, A Poincaré–Bendixson theorem for translation lines and applications to prime ends, Comment. Math. Helv. 94 (2019), 141–183.
  5. R.B. Walker, Periodicity and decomposability of basin boundaries with irrational maps on prime ends, Trans. Amer. Math. Soc. 324 (1991), 303–317.

Welington Cordeiro, Polish Academy of Sciences, Poland

BEYOND TOPOLOGICAL HYPERBOLICITY

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building A-3/A-4, room 103.

Abstract

We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. First of all, we will talk about some of these generalizations of expansivity and show some interesting examples. In particular, we will define entropy expansivity, \(N\)-expansivity and measure expansivity and we will give a overview about the recent results for these systems.

References

  1. A. Artigue, Dendritations of surfaces, Ergodic Theory Dynam. Systems 38 (2018), 2860-2912.
  2. A. Artigue, M.J. Pacífico, J. Vieitez, \(N\)-expansive homeomorphisms on surfaces, Communications in Contemporary Mathematics 19 (2017).
  3. R. Bowen, Entropy-Expansive Maps, Transactions of the American Mathematical Society 164 (1972), 323-331.
  4. B. Carvalho, W. Cordeiro, \(N\)-expansive homeomorphisms with the shadowing property, Journal of Differential Equations 261 (2016), 3734-3755.
  5. B. Carvalho, W. Cordeiro, Positively \(N\)-expansive homeomorphisms and the \(L\)-shadowing property, J. Dyn. Diff. Equat. 261 (2018), https://doi.org/10.1007/s10884-018-9698-3.
  6. W. Cordeiro, M. Denker, X. Zhang, On specification and measure expansiveness, Discrete and Continuous Dynamical Systems 37 (2017), 1941-1957.
  7. H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598.
  8. J. Li, R. Zhang, Levels of Generalized Expansiveness, Journal of Dynamics and Differential Equations, (2015), 1-18.
  9. R. Mañé, Expansive homeomorphisms and topological dimension, Trans, AMS 252 (1979), 313–319.
  10. C.A. Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst. 32(1) (2012), 293-301.

Udayan B. Darji, University of Louisville, USA

SOME APPLICATIONS OF LOCAL ENTROPY THEORY

Date: 2019-09-17 (Tuesday); Time: 16:55-17:15; Location: building A-3/A-4, room 103.

Abstract

In this talk we discuss some applications of local entropy theory, in particular results of Kerr-Li, to problems in topological dynamics and induced dynamics on the space of probability measures.

In the setting of topological dynamics, we discuss how local entropy theory can be used to show that in certain settings, the complexity of a dynamical system implies indecomposability in the inverse limit space of the dynamical system [2, 3], settling some old problems stated in [1].

A topological dynamical system \((X,f)\) induces natural dynamics on \(P(X)\), the space of probability measure on \(X\) defined by \(\tilde{f} :P(X) \rightarrow P(X)\) by \(\tilde{f} (\mu) = \mu f^{-1}\). A nontrivial and a remarkable result is that \(f\) has topological entropy zero if and only if \(\tilde{f}\) has measure zero [4]. Using techniques of [4], recently it was shown [5] that one can sharpen this result to null systems. We discuss how local entropy theory can be used to prove theorems of these types with ease.

References

  1. M. Barge, J. Martin, Chaos, periodicity, and snakelike continua, Trans. Amer. Math. Soc. 289(1) (1985), 355-365.
  2. C. Mouron, Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua, Proc. Amer. Math. Soc. 139(8) (2011), 2783-2791.
  3. U.B. Darji, H. Kato, Chaos and Indecomposability, Adv. Math. 304 (2017), 793–808.
  4. E. Glasner and B. Weiss, Quasi-factors of zero entropy systems, J. Amer. Math. Soc. 8(3) (1995), 665–686.
  5. Y. Qiao and X. Zhou, Zero sequence entropy and entropy dimension, Discrete and Continuous Dynam. Systems, vol 37, Number 1 (2017), no. 2, 435–448.

Matúš Dirbák, Matej Bel University, Slovakia

PRODUCT-MINIMAL SPACES

Joint work with Ľubomír Snoha and Vladimír Špitalský

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-3/A-4, room 103.

Abstract

We call a compact metric space \(Y\) product-minimal (respectively, homeo-product-minimal) if for every minimal system \((X,T)\) there is a continuous map (respectively, a homeomorphism) \(S\colon Y\to Y\) such that the product system \((X\times Y,T\times S)\) is minimal. Every homeo-product-minimal space is product-minimal and every product-minimal space is minimal, while the converse implications do not hold. In the talk we shall present examples of (homeo-)product-minimal spaces and list some operations, under which the class of all (homeo-)product-minimal spaces is closed.

References

  1. M. Dirbák, Ľ. Snoha, V. Špitalský, Minimal direct products, preprint (2019), 40 pp.

Sebastián Donoso, University of Chile, Chile

TOPOLOGICAL AND COMBINATORIAL PROPERTIES OF FINITE RANK MINIMAL SUBSHIFTS

Joint work with Fabien Durand, Alejandro Maass, and Samuel Petite

Date: 2019-09-16 (Monday); Time: 17:45-18:05; Location: building A-3/A-4, room 103.

Abstract

I will discuss topological and combinatorial properties of finite rank minimal systems, establishing a clear connection with the \(S\)-adic subshifts, under recognizability assumptions. I will also mention results concerning the asymptotic components of a finite rank subshift and show that there is a rank two minimal subshift with superlinear complexity. I will mention results concerning the automorphism group of a finite rank subshift and state some open questions.

References

  1. V. Berthé, W. Steiner, J. Thuswaldner and R. Yassawi, Recognizability for sequences of morphisms, Ergodic Theory Dynam. Systems, to appear.
  2. S. Donoso, F. Durand, A. Maass and S. Petite, From bounded rank to \(S\)-adic subshifts: complexity and automorphism groups, preprint 2019.
  3. T. Downarowicz and A. Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), 739-747.

Bartosz Frej, Wrocław University of Science and Technology, Poland

FACTORING GROUP SHIFTS ONTO THE FULL SHIFT

Joint work with Dawid Huczek

Date: 2019-09-17 (Tuesday); Time: 17:20-17:40; Location: building A-3/A-4, room 103.

Abstract

It is known that any subshift of finite type with the action of \(\mathbb{Z}\) and entropy greater or equal than \(\log n\) factors onto the full shift over \(n\) symbols (see [7] and [1] for the cases of equal and unequal entropy, respectively). Extending these results for actions of other groups has been difficult, and it is known that a factor map onto a full shift of equal entropy may not exist in this case (see [2]). Johnson and Madden showed in [6] that any SFT with the action of \(\mathbb{Z}^d\), which has entropy greater than \(\log n\) and satisfies an additional mixing condition (known as corner gluing), has an extension which is finite-to-one (hence of equal entropy) and maps onto the full shift over \(n\) symbols. This result was later improved by Desai in [4] and finally by Boyle, Pavlov and Schraudner in [3].

I will prove that in the case of actions of a countable amenable group, any strongly irreducible symbolic dynamical system with entropy greater than \(\log n\) has an equal-entropy symbolic extension which factors onto the full shift over \(n\) symbols. The construction uses tilings of amenable groups as presented in [5].

References

  1. M. Boyle, Lower entropy factors of sofic systems, Ergodic Theory Dynam. Systems 3(4) (1983), 541–557.
  2. M. Boyle, M. Schraudner, \(\mathbb{Z}^d\) shifts of finite type without equal entropy full shift factors, J. Differ. Equations Appl. 15 (2009), 47–52.
  3. M. Boyle, R. Pavlov, M. Schraudner, Multidimensional sofic shifts without separation and their factors, Trans. Amer. Math. Soc. 367, 5 (2015), 3371–3421.
  4. A. Desai, A class of \(\mathbb{Z}^d\) shifts of finite type which factors onto lowe entropy full shifts, Proc. Amer. Math. Soc. 27(8) (2009), 2613-2621.
  5. T. Downarowicz, D. Huczek, G. Zhang, Tilings of amenable groups, J. Reine Angew. Math 747 (2019), 277-298.
  6. A. Johnson, K. Madden, Factoring higher-dimensional shifts of finite type onto the full shift, Ergodic Theory Dynam. Systems 25 (2005), 811-822.
  7. B. Marcus, Factors and extensions of full shifts, Monatsh. Math. 88(3) (1979), 239-247.

Gabriel Fuhrmann, Imperial College London, UK

TAME IMPLIES REGULAR

Joint work with Eli Glasner, Tobias Jäger, and Christian Oertel

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building A-3/A-4, room 103.

Abstract

The last decade saw an increased interest in tame systems revealing their connections to different areas of mathematics like Banach spaces, substitutions and tilings, quasicrystals, cut and project schemes and even model theory and logic. A major breakthrough in the general understanding of tameness was achieved by Glasner's recent structural result for tame minimal systems [1]. One of its consequences is that a tame minimal dynamical system which has an invariant measure is almost automorphic, uniquely ergodic and measure-theoretically isomorphic to its maximal equicontinuous factor.

In this talk, we prove that tame minimal dynamical systems \((X,G)\) with an invariant measure are actually regularly almost automorphic, that is, they allow for a factor map \(\pi\) from \((X,G)\) to an equicontinuous system \((\mathbb T,G)\) such that almost every point in \(\mathbb T\) (with respect to the unique invariant measure on \(\mathbb T\)) has a unique preimage under \(\pi\), see [2].

References

  1. E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math. 211 (2018), 213-244.
  2. G. Fuhrmann, E. Glasner, T. Jäger, C. Oertel, Irregular model sets and tame dynamics, arXiv:1811.06283, (2018), 1-22.

Felipe García-Ramos, Autonomous University of San Luis Potosí, Mexico

TOPOLOGICAL MODELS OF KRONECKER SYSTEMS

Joint work with Tobias Jäger, Xiangdong Ye, and Dominik Kwietniak

Date: 2019-09-17 (Tuesday); Time: 10:40-11:00; Location: building A-3/A-4, room 103.

Abstract

I will talk about the range of behaviours of topological models of Kronecker systems (a.k.a. discrete spectrum systems) and loosely Kronecker systems (a.k.a. zero entropy loosely Bernoulli).

References

  1. F. García-Ramos, T. Jäger and X. Ye, Mean equicontinuity, almost automorphy and regularity, Preprint, 2019.
  2. F. García-Ramos, D. Kwietniak, Topological models of loosely Kronecker systems, Preprint, 2019.

Yonatan Gutman, Polish Academy of Sciences, Poland

A PROBABILISTIC TAKENS THEOREM

Joint work with Krzysztof Barański and Adam Śpiewak

Date: 2019-09-19 (Thursday); Time: 16:55-17:15; Location: building A-3/A-4, room 103.

Abstract

Let \(X \subset \mathbb{R}^N\) be a Borel set, \(\mu\) a Borel probability measure on \(X\) and \(T:X \to X\) a Lipschitz and injective map. Fix \(k \in \mathbb{N}\) greater than the (Hausdorff) dimension of \(X\) and assume that the set of \(p\)-periodic points has dimension smaller than \(p\) for \(p=1, \ldots, k-1\). We prove that for a typical polynomial perturbation \(\tilde{h}\) of a given Lipschitz map \(h : X \to \mathbb{R}\), the \(k\)-delay coordinate map \(x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))\) is injective on a set of full measure \(\mu\). This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required measurements from \(2\dim X\) to \(\dim X\) and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results and settle conjectures in the physics literature.

Jozef Kováč, Comenius University in Bratislava, Slovakia

DISTRIBUTIONAL CHAOS IN RANDOM DYNAMICAL SYSTEMS

Date: 2019-09-19 (Thursday); Time: 17:20-17:40; Location: building A-3/A-4, room 103.

Abstract

Consider two continuous interval maps \(f,g:[0,1]\to [0,1]\) and the random dynamical system given by \[ x_{n+1}= \begin{cases} f(x_n) & \text{ with probability } p,\\ g(x_n) & \text{ with probability } 1-p, \end{cases} \] where \(p\in (0,1)\). Distributional chaos for such systems was defined in [1]. We will discuss some of its properties (for example stability, types of distributional chaos, etc.).

References

  1. J. Kováč, K. Janková, Distributional chaos in random dynamical systems, J. Difference Equ. Appl. 25(4) (2019), 455–480.

Krystyna Kuperberg, Auburn University, USA

A MEASURE PRESERVING PL MODIFICATION OF THE JONES-YORKE BOUNDED ORBIT FLOW ON \(\mathbb{R}^3\)

Joint work with Jeffrey Ford

Date: 2019-09-19 (Thursday); Time: 16:20-16:50; Location: building A-3/A-4, room 103.

Abstract

We present a modification of the example by G. Stephen Jones and James A. Yorke of a smooth, bounded orbit, dynamical system on \(\mathbb{R}^3\). Our example is piecewise linear and measured. We use methods of piecewise linear dynamical system developed by Greg~Kuperberg, in particular his notion of a slanted suspension.

References

  1. G.S. Jones, J.A. Yorke, The existence and nonexistence of critical points in bounded flows, J. Differential Equations 6 (1969), 238-246.
  2. G. Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996), 70-97.

Dominik Kwietniak, Jagiellonian University in Kraków, Poland

ON THE ABUNDANCE OF (NON)TAME GROUP ACTIONS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building A-3/A-4, room 103.

Abstract

Tame dynamical systems were introduced by Köhler [3] in 1995. Tameness is a topological notion roughly corresponding to compactness appearing in the compactness vs weak mixing dichotomy that underlines the structure theory for measure preserving actions.

In recent years several authors developed the theory of tame systems revealing connections to other areas of mathematics like Banach spaces, circularly ordered systems, substitutions and tilings, quasicrystals, cut and project schemes and even model theory and logic.

During my talk, I will discuss results of [1], where we study tameness and nullness of regular almost automorphic \(G\)-actions utilising a generalised notion of semi-cocycle extensions. In particular, we show that every ergodic equicontinuous \(G\)-action on a compact metric space admits a regular almost automorphic extension which is non-tame as well as tame but non-null extension. In some sense, this complements a recent result of Glasner [2]. We prove that such examples appear in well-studied families of group actions including Delone dynamical systems and symbolic systems (including Toeplitz flows over arbitrary \(G\)-odometers).

References

  1. G. Fuhrmann, D. Kwietniak, On tameness of almost automorphic dynamical systems for general groups, preprint, arXiv:1902.10780 [math.DS] (2019).
  2. E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math.211(1) (2018), 213-244.
  3. A. Köhler, Enveloping semigroups for flows, Proc. Roy. Irish Acad. Sect. A 95(2) (1995), 179-191.

Jian Li, Shantou University, China

RECENT DEVELOPMENTS ON MEAN EQUICONTINUITY AND MEAN SENSITIVITY

Date: 2019-09-17 (Tuesday); Time: 11:05-11:25; Location: building A-3/A-4, room 103.

Abstract

In this talk we will discuss recent developments on mean equicontinuity and mean sensitivity both on topological and measure-theoretic settings, based on the following three papers and related works.

References

  1. J. Li, S. Tu, X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems 35 (2015), 2587-2612.
  2. J. Li, Measure-theoretic sensitivity via finite partitions, Nonlinearity 29 (2016), 2133-2144.
  3. W. Huang, J. Li, J. Thouvenot, L. Xu, X. Ye, Bounded complexity, mean equicontinuity and discrete spectrum, Ergodic Theory and Dynamical Systems, preprint, arXiv:1806.02980.

Michał Misiurewicz, Indiana University - Purdue University Indianapolis, USA

RENORMALIZATION TOWERS AND THEIR FORCING

Joint work with Alexander Blokh

Date: 2019-09-16 (Monday); Time: 16:20-16:50; Location: building A-3/A-4, room 103.

Abstract

Over half a century ago, Sharkovsky proved his theorem about periodsof periodic orbits of continuous interval maps. Existence of someperiods force existence of some other periods, and the orderingobtained in such a way is linear. Later, people noticed that insteadof looking at periods, one can take into account permutations.Unfortunately, this gives only a partial order, which is verycomplicated and impossible to describe in simple terms. We propose themiddle ground: to look at the block structures of permutations (theycan be also understood in terms of renormalizations). This is a finerclassification of periodic orbits than just by periods, but stillresults in a linear ordering.

Piotr Oprocha, AGH University of Science and Technology, Poland

ON THE ENTROPY CONJECTURE OF MARCY BARGE

Joint work with Jan Boroński and Jernej Činč

Date: 2019-09-17 (Tuesday); Time: 16:20-16:50; Location: building A-3/A-4, room 103.

Abstract

I shall discuss a positive solution to the following problem, obtained in a joint work with J. Boroński and J. Činč.

Question (M. Barge, 1989 [8]) Does there exist, for every \(r\in [0,\infty]\), a pseudo-arc homeomorphism whose topological entropy is \(r\)?

Until now all known pseudo-arc homeomorphisms have had entropy \(0\) or \(\infty\). Recall that the pseudo-arc is a compact and connected space (continuum) first constructed by Knaster in 1922 [6]. It can be seen as a pathological fractal. According to the most recent characterization [5] it is topologically the only, other than the arc, continuum in the plane homeomorphic to each of its proper subcontinua. The pseudo-arc is homogeneous [2] and played a crucial role in the classification of homogeneous planar compacta [4]. Lewis showed that for any \(n\) the pseudo-arc admits a period \(n\) homeomorphism that extends to a rotation of the plane, and that any \(P\)-adic Cantor group action acts effectively on the pseudo-arc [7] (see also [10]). We adapt Lewis' inverse limit constructions, by combining them with a Denjoy-Rees scheme [1] (see also [9], [3]). The positive entropy homeomorphisms that we obtain are periodic point free, except for a unique fixed point.

I am going to present various results related to the problem, to conclude with a discussion of its solution.

References

  1. F. Béguin, S. Crovisier, F. Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique, Ann. Sci. École Norm. Sup. 40 (2007), 251-308.
  2. R.H. Bing, homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742.
  3. J.P. Boroński, J. Kennedy, X. Liu, P. Oprocha, Minimal noninvertible maps on the pseudocircle, arXiv:1810.07688.
  4. L.C. Hoehn, L.G. Oversteegen, A complete classification of homogeneous plane continua, Acta Math. 216 (2016), 177-216.
  5. L.C. Hoehn, L.G. Oversteegen, A complete classification of hereditarily equivalent plane continua, arXiv:1812.08846.
  6. B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math. 3 (1922), 247-286.
  7. W. Lewis, Periodic homeomorphisms of chainable continua, Fund. Math. 117 (1983), 81-84.
  8. W. Lewis, Continuum theory and dynamics problems, Continuum theory and dynamical systems (Arcata, CA, 1989), 99–101, Contemp. Math., 117, Amer. Math. Soc., Providence, RI, 1991.
  9. M. Rees, A minimal positive entropy homeomorphism of the 2-torus, J. London Math. Soc. 2 (1981), 537-550.
  10. J. Toledo, Inducible periodic homeomorphisms of tree-like continua, Trans. Amer. Math. Soc. 282 (1984), 77–108.

Ronnie Pavlov, University of Denver, USA

NON-UNIFORM SPECIFICATION PROPERTIES ON SUBSHIFTS

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building A-3/A-4, room 103.

Abstract

A celebrated result of Bowen implies uniqueness of equilibrium state for certain potentials on expansive systems with the specification property. In the setting of symbolic dynamics, this property is equivalent to the existence of a constant \(N\) such that any two \(n\)-letter words \(v,w\) in the language can be combined into a new word in the language given a gap between them of length at least \(N\). Several weakenings of specification have seen recent activity, among them non-uniform specification, in which one allows the gap to have size controlled by an increasing function \(f(n)\). I will summarize some known results about non-uniform specification, including a provable threshold on \(f(n)\) below which one can prove generalizations of Bowen's result on uniqueness of equilibrium state.

References

  1. R. Pavlov, On non-uniform specification and uniqueness of the equilibrium state in expansive systems, Nonlinearity 32 (2019), 2441-2466.
  2. R. Pavlov, On intrinsic ergodicity and weakenings of the specification property, Adv. Math. 295 (2016), 250-270.

Vojtěch Pravec, Silesian University in Opava, Czech Republic

REMARKS ON DEFINITIONS OF PERIODIC POINTS FOR NONAUTONOMOUS DYNAMICAL SYSTEM

Date: 2019-09-16 (Monday); Time: 16:55-17:15; Location: building A-3/A-4, room 103.

Abstract

Let \((X,f_{1,\infty})\) be a nonautonomous dynamical system. In this talk we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new definition of asymptotic periodicity. This definition is not only very natural but also resistant to changes of a beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for uniformly convergent nonautonomous systems topological transitivity together with dense set of asymptotically periodic points imply sensitivity. We also show that even for uniformly convergent systems the nonautonomous analog of Sharkovsky’s Theorem is not valid for most definitions of periodic points.

References

  1. J.S. Cánovas, Li-Yorke chaos in a class of nonautonomous discrete systems, Journal of Difference Equations and Applications 17 (2011), 479–486.
  2. A. Miralles, M. Murillo-Arcila, M. Sanchis, Sensitive dependence for nonautonomous disrcete dynamical systems, Journal of Mathematical Analysis and Applications 463 (2018), 268–275.
  3. Y. Shi, G. Chen, Chaos of time-varying discrete dynamical systems, Journal of Difference Equations and Applications 15 (2009), 429–449.

Ľubomír Snoha, Matej Bel University, Slovakia

ON THE PROBLEM OF THE TOPOLOGICAL CLASSIFICATION OF MINIMAL SETS

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-3/A-4, room 103.

Abstract

One of the open problems in topological dynamics is the problem of the topological classification of minimal spaces/sets.

A dynamical system \((X,f)\) given by a topological space \(X\) and a continuous map \(f: X\to X\) is called minimal if all forward orbits are dense. A nonempty closed set \(M\subseteq X\) with \(f(M)\subseteq M\) is a minimal set for the system \((X,f)\) or for the map \(f\), if \((M, f|_M)\) is a minimal system. Thus, a system \((X,f)\) is minimal if and only if \(X\) is a minimal set. In every compact system there are minimal sets. A space \(X\) is said to be minimal if it admits a minimal (in general noninvertible) map.

The classification problem has two parts:

(1) Which spaces \(X\) are minimal and which are not?

(2) Given a space \(X\), consider all possible continuous selfmaps of this space and their minimal sets. Describe these sets topologically (find their full topological characterization).

(An illustrating example to (2): The minimal sets in \(I=[0,1]\) are nonempty finite sets and Cantor sets, meaning that if \(f\colon I\to I\) is continuous and \(M\) is a minimal set of \(f\), then \(M\) is either nonempty finite or Cantor and, conversely, if \(M \subseteq I\) is nonempty finite or Cantor, then there is a continuous map \(f\colon I\to I\) such that \(M\) is a minimal set of \(f\).)

In the talk we give a survey of some results on the classification problem.

Gabriel Soler López, Technical University of Cartagena, Spain

MINIMAL INTERVAL EXCHANGE TRANSFORMATIONS AND MINIMAL SURFACES

Joint work with José Ginés Espín Buendía, Antonio Linero Bas ,and Daniel Peralta Salas

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-3/A-4, room 103.

Abstract

An interval exchange transformation of \(n\)-interval, abbreviately \(n\)-IET, is an injective map \(T:D\subset[0,1]\to [0,1]\) such that:

\(\bullet\) \(D\) is the union of \(n\) pairwise disjoint open intervals, \(D=\cup_{i=1}^n I_i\), with \(I_i=]a_i,a_{i+1}[\), \(a_1=0\), \(a_{n+1}=1\) and \(n\geq 2\);

\(\bullet\) \(T|_{I_i}\) is a map of constant slope equals to \(1\) or \(-1\);

If \(T\) reverses the orientation in the interval set \(\mathcal{F}=\{I_{f_1},I_{f_2},\dots, I_{f_k}\}\) (the slope is \(-1\) in these intervals) for some \(1\le f_j\le n\) then we stress it by saying that \(T\) is an interval exchange transformation of \(n\)-intervals with \(k\)-flips or an (n,k)-IET; otherwise we say that \(T\) is an interval exchange transformation of \(n\)-intervals without flips or an oriented interval exchange transformation of \(n\)-intervals.

The point \(a_i\) is said to be a false discontinuity if \(\lim_{x\to a_i^+} T(x)=\lim_{x\to a_i^-} T(x)\). We will say that \(T\) is a proper \((n,k)\)-IET when it has not false discontinuities.

Let \(x\in [0,1]\) then the orbit of \(x\) under \(T\) is the set: \[\mathcal{O}_T(x)=\{T^n(x): n \textrm{ is an integer and } T^n(x) \textrm{ makes sense} \}.\] \(T\) is said to be minimal if for any \(x\in[0,1]\) then \(\mathcal{O}_T(x)\) is dense in \([0,1].\)

Let \(\mu\) denote the standard Lebesgue measure on \([0,1]\). It is easy to see that \(\mu\) (and any of its multiples) is an invariant measure for any interval exchange transformation. \(T\) will be said to be uniquely ergodic if it does not admit other invariant measures. We will present some progress in the theory of interval exchange transformations with flips. In particular we will pay attention to the Main Theorem in [2] which assures the existence of minimal, uniquely ergodic, proper \((n,k)\)-IET's for any \(n,k\in\mathbb{N}\) with \(n\ge 4\) and \(1\le k\le n\). Also it will be explained the study non-oriented surfaces admitting minimal flows made in [1] and the work in progress to build minimal non uniquely ergodic flipped IET's.

Acknowledgements

This talk has been partially supported by the grant number MTM 2017-84079-P from Ministerio de Ciencia Innovación y Universidades (Spain)

References

  1. J.G. Espín, D. Peralta, and G. Soler, Existence of minimal flows on nonorientable surface, Discrete and Continuous Dynamical Systems. Series A, 37 (2017), 4191–4211.
  2. A. Linero, G. Soler López, Minimal interval exchange transformations with flips, Ergodic Theory Dynam. Systems. 38 (2018), 3101–3144.

Andy Zucker, Université Paris Diderot, France

BERNOULLI DISJOINTNESS

Joint work with Eli Glasner, Todor Tsankov, and Benjamin Weiss

Date: 2019-09-16 (Monday); Time: 17:20-17:40; Location: building A-3/A-4, room 103.

Abstract

We consider the concept of disjointness for topological dynamical systems, introduced by Furstenberg. We show that for every discrete group, every minimal flow is disjoint from the Bernoulli shift. We apply this to give a negative answer to the Ellis problem for all such groups. For countable groups, we show in addition that there exists a continuum-sized family of mutually disjoint free minimal systems. Using this, we can identify the underlying space of the universal minimal flow of every countable group, generalizing results of Balcar-Blaszczyk and Turek. In the course of the proof, we also show that every countable ICC group admits a free minimal proximal flow, answering a question of Frisch, Tamuz, and Vahidi Ferdowsi.

Matthieu Astorg, Université d'Orléans, France

WANDERING DOMAINS ARISING WITH FROM LAVAURS MAPS WITH SIEGEL DISKS

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building A-3/A-4, room 105.

Abstract

A famous theorem of Sullivan asserts that polynomials in one complex variables have no wandering Fatou components. On the other hand, in a joint work with Buff, Dujardin, Peters and Raissy, we constructed the first examples of polynomial maps in two complex variables having such components. The construction relies on parabolic implosion, and involves the dynamics of non-autonomous perturbations of a Lavaurs map with an attracting fixed point. In this talk, we will present a more recent work with Boc-Thaler and Peters, in which we classify the local dynamics in the case where the Lavaurs map has a Siegel fixed point. In particular, we prove that wandering domains may also arise in that setting. Time permitting, we will introduce the notion of parabolic curves and how their existence simplifies the proof

Krzysztof Barański, University of Warsaw, Poland

SLOW ESCAPING POINTS FOR TRANSCENDENTAL MAPS

Joint work with Bogusława Karpińska

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building A-3/A-4, room 105.

Abstract

Let \(f \colon \mathbb{C} \to \mathbb{C}\) be a transcendental entire map. The set \[ I(f) = \{z \in \mathbb{C} : f^n(z) \to \infty\} \] is called the escaping set of \(f\). In relation with the papers [1, 2], we study the dimension of the sets of points in \(I(f)\) which escape to infinity in a given rate.

References

  1. W. Bergweiler, J. Peter, Escape rate and Hausdorff measure for entire functions, Math. Z. 274 (2013), 551-572.
  2. D.J. Sixsmith, Dimensions of slowly escaping sets and annular itineraries for exponential functions, Ergodic Theory Dynam. Systems 36 (2016), 2273-2292.

Fabrizio Bianchi, CNRS & Université de Lille, France

LATTES MAPS AND THE HAUSDORFF DIMENSION OF THE BIFURCATION LOCUS

Joint work with François Berteloot

Date: 2019-09-20 (Friday); Time: 12:05-12:25; Location: building A-3/A-4, room 105.

Abstract

Given a holomorphic family of endomorphisms of \(P^k (C)\), the bifurcation locus is the set of parameters of instability for the Julia set, the support of the measure of maximal entropy. This locus coincides with the non-harmonicity locus of the Lyapunov function, the sum of the Lyapunov exponents. Lattes maps can be characterised as the minima of the Lyapunov function, and thus lie in the bifurcation locus. We will prove that near isolated Lattes maps the Hausdorff dimension of the bifurcation locus is maximal in any direction.

Alexander Blokh, University of Alabama at Birmingham, USA

LOCATION OF SIEGEL CAPTURE POLYNOMIALS IN PARAMETER SPACES OF CUBIC POLYNOMIALS

Joint work with Arnaud Chéritat, Lex Oversteegen, and Vladlen Timorin

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building A-3/A-4, room 105.

Abstract

Consider cubic polynomials with a Siegel disk containing an eventual image of a critical point and call them IS-capture polynomials ("IS" stands for Invariant Siegel). We study the location of IS-capture polynomials in the parameter space of all cubic polynomials and show that any IS-capture polynomial belongs to the boundary of a unique bounded hyperbolic component determined by the rational lamination of the map.

Trevor Clark, Imperial College London, UK

CONJUGACY CLASSES OF REAL ANALYTIC MAPPINGS

Joint work with Sebastian van Strien

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building A-3/A-4, room 105.

Abstract

I will discuss recent results on the manifold structure of the topological conjugacy classes of real-analytic mappings. These results are based on the construction of a “pruned polynomial-like mapping" associated to a real mapping. This gives us an ''external structure'' for a real-analytic mapping.

Romain Dujardin, Sorbonne Université, France

DYNAMICS OF UNIFORMLY HYPERBOLIC HÉNON MAPS

Joint work with Eric Bedford and Misha Lyubich

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-3/A-4, room 105.

Abstract

In this talk I will review some new recent results on the dynamics of uniformly hyperbolic Hénon maps. Topics covered will include the \(J=J^*\) problem, some new geometric and topological criteria for hyperbolicity, and the topological structure of the Julia set for hyperbolic maps.

Christophe Dupont, Université de Rennes 1, France

DYNAMICS OF FIBERED ENDOMORPHISMS OF \(\mathbb C \mathbb P(2)\)

Joint work with Johan Taflin

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building A-3/A-4, room 105.

Abstract

The talk concerns the endomorphisms of \(\mathbb C \mathbb P(2)\) preserving a pencil of lines, those maps generalize the polynomial skew products of \(\mathbb C^2\) studied by Jonsson. We show that the equilibrium measure of those endomorphisms decomposes (Fubini's formula relative to the invariant pencil) and we study its Lyapunov exponents. One of them is equal to the exponent of the rational map acting on the pencil. We provide for the other one a formula involving a relative Green function and the critical set. In particular, that exponent is larger than the logarithm of the degree of the endomorphism. This is a joint work with Johan Taflin.

Núria Fagella, University of Barcelona, Spain

WANDERING DOMAINS IN TRANSCENDENTAL FUNCTIONS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building A-3/A-4, room 105.

Abstract

Wandering domains are Fatou components that only exist in the transcendental setting. Although important progress has taken place in the past few years, there are still many open questions. In this talk I will review the state of the art on the existence and classification of wandering domains, and their relation with the singularities of the inverse map. I shall present some recent results on these topics.

Igors Gorbovickis, Jacobs University, Germany

ON RENORMALIZATION OF CRITICAL CIRCLE MAPS WITH NONINTEGER EXPONENTS

Date: 2019-09-19 (Thursday); Time: 10:40-11:00; Location: building A-3/A-4, room 105.

Abstract

We discuss some results and ongoing developments in the study of renormalization of critical circle maps with non-integer critical exponents sufficiently close to odd integers.

References

  1. I. Gorbovickis, M. Yampolsky, Rigidity, universality,and hyperbolicity of renormalization for critical circle maps with non-integer exponents, to appear in Ergodic Theory Dynam. Systems.
  2. I. Gorbovickis, M. Yampolsky, Renormalization for unimodal maps with non-integer exponents, Arnold Math. Journal 4(2) (2018), 179-191.

Xavier Jarque, Universitat de Barcelona & IMUB, Catalonia

UNIVALENT WANDERING DOMAINS IN THE EREMENKO-LYUBICH CLASS

Joint work with Núria Fagella and Kirill Lazebnik

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building A-3/A-4, room 105.

Abstract

We use the Folding Theorem of [1] to construct an entire function \(f\) in class \(\mathcal{B}\) and a wandering domain \(U\) of \(f\) such that \(f\) restricted to \(f^n(U)\) is univalent, for all \(n\geq 0\). The components of the wandering orbit are bounded and surrounded by the postcritical set.

References

  1. C. Bishop, Constructing entire functions by quasiconformal folding, Acta Mathematica 214(1) (2015), 1-60.

Zhuchao Ji, Sorbonne Université, France

NON-UNIFORM HYPERBOLICITY IN POLYNOMIAL SKEW PRODUCTS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-3/A-4, room 105.

Abstract

The dynamics of Topological Collet-Eckmann rational maps on Riemann sphere are well understood, due to the work of Przytycki, Rivera-Letelier and Smirnov [1, 2]. In this talk we study the dynamics of polynomial skew products of \(\mathbb{C}^2\). Let \(f\) be a polynomial skew products with an attracting invariant line \(L\), such that \(f\) restricted on \(L\) satisfies Topological Collet-Eckmann condition and a Weak Regularity condition. We show that the the Fatou set of \(f\) in the basin of \(L\) equals to the union of the basins of attracting cycles, and the Julia set of \(f\) in the basin of \(L\) has Lebesgue measure zero. As a consequence there are no wandering Fatou components in the basin of \(L\) (we remark that for some polynomial skew products with a parabolic invariant line \(L\), there can exist a wandering Fatou component in the basin of \(L\) [3, 4]).

References

  1. F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Inventiones mathematicae 151 (2003), 29–63.
  2. F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Annales Scientifiques de l’École Normale Superieure 40 (2007), 135–178.
  3. M. Astorg, X. Buff, R. Dujardin, H. Peters and J. Raissy, A two-dimensional polynomial mapping with a wandering Fatou component, Annals of mathematics 184 (2016), 263– 313.
  4. M. Astorg, L. Boc-Thaler and H. Peters, Wandering domains arising from Lavaurs maps with Siegel disks, arXiv preprint arXiv:1907.04140 (2019).
  5. Z. Ji, Non-uniform hyperbolicity in polynomial skew products, In preparation.

Kirill Lazebnik, California Institute of Technology, USA

UNIVALENT POLYNOMIALS AND HUBBARD TREES

Joint work with Nikolai G. Makarov and Sabyasachi Mukherjee

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building A-3/A-4, room 105.

Abstract

We study the space of ''external polynomials'' \[ \Sigma_d^* := \left\{ f(z)= z+\frac{a_1}{z} + \cdots +\frac{a_d}{z^d} : a_d=-\frac{1}{d}\textrm{ and } f|_{\hat{\mathbb{C}}\setminus\overline{\mathbb{D}}} \textrm{ is conformal}\right\}. \] It is proven that a simple class of combinatorial objects (bi-angled trees) classify those \(f\in\Sigma_d^*\) with the property that \(f(\mathbb{T})\) has the maximal number \(d-2\) of double points. We discuss a surprising connection with the class of anti-holomorphic polynomials of degree \(d\) with \(d-1\) distinct, fixed critical points and their associated Hubbard trees.

References

  1. Lazebnik, Kirill, Makarov, Nikolai, Mukherjee, Sabyasachi, Univalent Polynomials and Hubbard Trees, arXiv, 2019.

David Martí-Pete, Polish Academy of Sciences, Poland

FINGERS IN THE PARAMETER SPACE OF THE COMPLEX STANDARD FAMILY

Joint work with Mitsuhiro Shishikura

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building A-3/A-4, room 105.

Abstract

We investigate the parameter space of the complex standard family \[F_{\alpha,\beta}(z)=z+\alpha+\beta \sin z,\] where the parameter \(0<\beta\ll 1\) is considered to be fixed and the bifurcation is studied with respect to the parameter \(\alpha\in\mathbb{C}\). This two parameter family of entire functions are lifts of holomorphic self-maps of \(\mathbb{C}^*\) that arise as the complexification of the Arnol'd standard family of circle maps. In the real axis of the \(\alpha\)-parameter plane one can observe the so-called Arnold tongues, given by the real parameters \((\alpha, \beta)\) such that \(F_{\alpha,\beta}\) has a constant rotation number. Their complex extension contain some finger-like structures which were observed for the first time by Fagella in her PhD thesis [1] that increase in number as \(\beta\to 0\). We study the qualitative and quantitative aspects of such fingers via parabolic bifurcation. In particular, we show that for every \(0<\beta\ll 1\) the number of fingers is finite and give an estimate of this quantity as \(\beta \to 0\). This is a very general capture phenomenon that appears in the parameter spaces of many families of holomorphic functions with more than one critical point.

References

  1. N. Fagella, Dynamics of the complex standard family, J. Math. Anal. Appl. 229(1) (1999), 1-31.

John Mayer, University of Alabama at Birmingham, USA

CRITICAL PORTRAITS, SIBLING PORTRAITS, THE CENTRAL STRIP, AND NEVER CLOSE SIDES OF POLYGONS IN LAMINATIONS

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building A-3/A-4, room 105.

Abstract

Laminations of the unit disk were introduced by William Thurston as a topological/combinatorial vehicle for understanding the (connected) Julia sets of polynomials, and, in particular, the parameter space of quadratic polynomials. Though the problem that Thurston was interested in has not been solved, the local connectedness of the Mandelbrot set (the analytic parameter space of quadratic polynomials), his excursion into laminations eventually gave birth to laminations as a way of understanding higher degree polynomials and their corresponding laminations. Much work has been done for cubic polynomials and their parameter spaces (analytic and laminational). In this talk we will decribe some work in progress on understanding phenomena that can occur with laminations, and consequently with Julia sets (maybe), of higher degree, \(d\ge 3.\) In particular, we are interested in laminational phenomena that cannot occur for \(d=2\), but can occur for \(d=3,\) cannot occur for \(d\le 3,\) but can occur for \(d=4,\) and so on. The topics mentioned in the title are on the route of discovery.

References

  1. A. Blokh, D. Mimbs, L. Oversteegen, K. Valkenburg, Laminations in the language of leaves, Trans. Amer. Math. Soc. 365 (2013), 5367-5391.
  2. A. Blokh, J. Malaugh, J. Mayer, L. Oversteegen, D. Parris, Rotational subsets of the circle under \(z^n\), Topology and Its Appl. 153 (2006), 1540-1570.
  3. D. Childers, Wandering polygons and recurrent critical leaves, Ergodic Theory Dynam. Systems 27(1) (2007), 87-107.
  4. D. Cosper, J. Houghton, J. Mayer, L. Mernik, J. Olson, Central Strips of sibling leaves in laminations of the unit disk, Topology Proc. 48 (2016), 69-100.
  5. J. Mayer, L. Mernik, Central Strip Portraits, preprint (2017).

Daniel Nicks, University of Nottingham, UK

THE ITERATED MINIMUM MODULUS AND EREMENKO’S CONJECTURE

Joint work with Phil Rippon and Gwyneth Stallard

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building A-3/A-4, room 105.

Abstract

Eremenko has conjectured that for any transcendental entire function \(f\), the escaping set \(I(f):= \{ z : f^n(z)\to\infty \mbox{ as } n\to\infty\}\) is connected. This talk will focus on real entire functions of finite order with only real zeroes. We show that Eremenko's conjecture holds for such a function \(f\) (and in fact \(I(f)\) has a "spider's web" structure) if there exists \(r>0\) such that the iterated minimum modulus \(m^n(r)\to\infty\) as \(n\to\infty\). Here \(m(r):=\min_{|z|=r}|f(z)|\). We will briefly discuss examples of families of functions for which this minimum modulus condition does, and does not, hold.

Lex Oversteegen, University of Alabama at Birmingham, USA

SLICES OF THE PARAMETER SPACE OF CUBIC POLYNOMIALS

Joint work with Alexander Blokh and Vladlen Timorin

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building A-3/A-4, room 105.

Abstract

In this talk we consider slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the main cubioid in this parameter space. The main cubioid is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials \(z^2+c\) for \(c\) in the filled main cardioid.

Jasmin Raissy, Université Paul Sabatier, France

A DYNAMICAL RUNGE EMBEDDING OF \(\mathbb{C}\times\mathbb{C}^*\) IN \(\mathbb{C}^2\)

Joint work with Filippo Bracci and Berit Stensønes

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building A-3/A-4, room 105.

Abstract

In this talk, I will present the construction of a family of automorphisms of \(\mathbb{C}^2\) having an invariant, non-recurrent Fatou component biholomorphic to \(\mathbb{C}\times \mathbb{C}^*\) and which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Such component is obtained by globalizing, thanks to a result of Forstneric, a local construction, which allows to create a global basin of attraction for an automorphism, and a Fatou coordinate on it. Such Fatou coordinate is a fiber bundle map on \(\mathbb{C}\), whose fiber is \(\mathbb{C}^*\), forcing the global basin to be biholomorphic to \(\mathbb{C}\times\mathbb{C}^*\). The most subtle point is to show that such a basin is indeed a Fatou component. This is done exploiting Pöschel’s results about existence of local Siegel discs and suitable estimates for the Kobayashi distance. This construction gives an example of a Runge embedding of \(\mathbb{C}\times \mathbb{C}^*\) in \(\mathbb{C}^2\), since attracting Fatou components are Runge domains.

References

  1. F. Bracci, J. Raissy, B. Stensønes, Automorphisms of \(\mathbb{C}^k\) with an invariant non-recurrent attracting Fatou component biholomorphic to \(\mathbb{C}\times(\mathbb{C}^*)^{k-1}\), to appear in Journal of the European Mathematical Society, www.ems-ph.org/journals/forthcoming.php?jrn=jems, arXiv:1703.08423.

Philip Rippon, Open University, UK

CONSTRUCTING BOUNDED SIMPLY CONNECTED WANDERING DOMAINS WITH PRESCRIBED DYNAMICS

Joint work with Anna Miriam Benini, Vasiliki Evdoridou, Nuria Fagella, and Gwyneth Stallard

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building A-3/A-4, room 105.

Abstract

We give a new general technique for constructing transcendental entire functions with bounded simply connected wandering domains, which allows us to prescribe the type of long term dynamics within the wandering domains. In some cases we can show that the wandering domains have Jordan curve boundaries.

Dierk Schleicher, Aix Marseille Université, France

THE RATIONAL RIGIDITY PRINCIPLE: TOWARDS RIGIDITY OF RATIONAL MAPS

Joint work with Kostiantyn Drach

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building A-3/A-4, room 105.

Abstract

Rigidity is one of the key goals in holomorphic dynamics; one way to phrase it is to say that any two maps can be distinguished in combinatorial terms. There are deep results and remarkable progress especially about polynomial maps. Much less is known about non-polynomial rational maps. We present recent progress, in joint work with Kostiantyn Drach, in establishing rigidity of a large class of rational maps, in particular Newton maps of polynomials. Similarly, we prove local connectivity of the corresponding Julia sets.

Nikita Selinger, University of Alabama at Birmingham, USA

PACMAN RENORMALIZATION

Joint work with Dima Dudko and Misha Lyubich

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building A-3/A-4, room 105.

Abstract

In a joint work with Misha Lyubich and Dima Dudko, we develop a theory of Pacman Renormalization inspired by earlier surgery construction by Branner and Douady. We show the hyperbolicity of periodic points of this renormalization with one-dimensional unstable manifold. This yield multiple consequences such as scaling law for centers of hyperbolic components attached to the main cardioid of the Mandelbrot set and local stability of certain Siegel discs.

Gwyneth Stallard, Open University, UK

CLASSIFYING SIMPLY CONNECTED WANDERING DOMAINS

Joint work with Anna Miriam Benini, Vasiliki Evdoridou, Nuria Fagella, and Phil Rippon

Date: 2019-09-17 (Tuesday); Time: 12:05-12:25; Location: building A-3/A-4, room 105.

Abstract

For rational functions, a classification of periodic Fatou components, with a detailed description of the dynamical behaviour inside each of the four possible types, was given around 100 years ago. For transcendental entire functions, there is an additional class known as Baker domains that are now well understood, and many examples of wandering domains, which cannot occur for rational functions. Although there is now a detailed description of the dynamical behaviour inside multiply connected wandering domains, there has been no systematic study of simply connected wandering domains. We show that there is in fact a wealth of possibilities for such domains and give a new classification into nine different types in terms of the hyperbolic distance between iterates and by whether orbits approach the boundaries of the domains. We give a new general technique for constructing bounded simply connected wandering domains which can be used to show that all nine types are realisable.

Johan Taflin, Université de Bourgogne, France

REGULARITY OF ATTRACTING CURRENTS

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-3/A-4, room 105.

Abstract

To each attractor of an endomorphism \(f\) of \(\mathbb{CP}^k\) it is possible to associate an analytic object called an attracting current. It can be used to obtain several information on the attractor and in this talk, I will explain how a weak form of regularity of this current is related to the dynamics of \(f\) on the attractor.

Vladlen Timorin, National Research University Higher School of Economics, Russia

INVARIANT SPANNING TREES FOR QUADRATIC RATIONAL MAPS

Joint work with Anastasia Shepelevtseva

Date: 2019-09-16 (Monday); Time: 12:05-12:25; Location: building A-3/A-4, room 105.

Abstract

A theorem of W. Thurston (sometimes called the fundamental theorem of complex dynamics) opens a door for algebraic, topological and combinatorial methods into dynamics of rational maps on the Riemann sphere. We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees. This procedure uses bisets over the fundamental group of a punctured sphere. We also introduce a new combinatorial invariant of Thurston classes - the ivy graph representing the pullback relation on (isotopy classes of) spanning trees.

Liz Vivas, Ohio State University, USA

NON-AUTONOMOUS PARABOLIC BIFURCATION

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building A-3/A-4, room 105.

Abstract

Let \(f(z) = z+z^2+O(z^3)\) and \(f_\epsilon(z) = f(z) + \epsilon^2\). A classical result in parabolic bifurcation [1, 2] in one complex variable is the following: if \(N-\frac{\pi}{\epsilon}\to 0\) we obtain \((f_\epsilon)^{N} \to \mathcal{L}_f\), where \(\mathcal{L}_f\) is the Lavaurs map of \(f\). In this paper we study a non-autonomous parabolic bifurcation. We focus on the case of \(f_0(z)=\frac{z}{1-z}\). Given a sequence \(\{\epsilon_i\}_{1\leq i\leq N}\), we denote \(f_n(z) = f_0(z) + \epsilon_n^2\). We give sufficient and necessary conditions on the sequence \(\{\epsilon_i\}\) that imply that \(f_{N}\circ\ldots f_{1} \to \textrm{Id}\) (the Lavaurs map of \(f_0\)). We apply our results to prove parabolic bifurcation phenomenon in two dimensions for some class of maps.

References

  1. A.Douady, Does a Julia set depend continuously on the polynomial?, Proc. Sympos. Appl. Math. 49 (1994), 91-138.
  2. P. Lavaurs, Systemes dynamiques holomorphes: explosion de points périodiques paraboliques, PhD thesis, Paris 11, 1989.

Jonguk Yang, Stony Brook University, USA

RECURRENCE OF ONE-SIDED SEQUENCES UNDER SHIFT

Date: 2019-09-19 (Thursday); Time: 11:05-11:25; Location: building A-3/A-4, room 105.

Abstract

Consider the shift map acting on the space of one-sided sequences. Under this dynamics, a sequence exhibits one of three types of recurrence: non-recurrence, reluctant recurrence, or persistent recurrence. However, for a given arbitrary sequence, it can be difficult to determine which of these three possibilities will occur. To solve this problem, we introduce an algebraic structure on sequences called filtration that enables us to count recurrences efficiently. This then leads to the characterization of the persistent recurrence property as a kind of infinite renormalizability of the shift map.

Jon Aaronson, Tel Aviv University, Israel

ON MIXING PROPERTIES OF INFINITE MEASURE PRESERVING TRANSFORMATIONS

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building A-4, room 106.

Abstract

I'll review the Hopf-Krickeberg mixing property with examples and discuss related ergodic properties such as rational weak mixing.

Tim Austin, University of California, Los Angeles, USA

RECENT PROGRESS ON STRUCTURE AND CLASSIFICATION IN ERGODIC THEORY

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

A basic formulation of the 'classification problem' asks for criteria to determine when two ergodic measure-preserving systems are isomorphic. It goes back to the foundational work of von Neuman and Halmos.

In more recent decades, this problem has evolved from 'pure' ergodic theory into a point of overlap with descriptive set theory. Culminating in work of Foreman, Rudolph and Weiss, this connection has shown the impossibility of any reasonable such classification. More recently, Foreman and Weiss have also shown that the important restriction of the classification problem to 'classical systems' - that is, smooth, volume-preserving maps of compact manifolds - is equally intractable.

However, some rich positive results are available in the direction of partial structure. These have the flavour that, for all ergodic measure-preserving systems, some 'soft' feature can be turned into a factor map or isomorphism to another system of a special kind. For instance, rotations on compact Abelian groups account for any failure of weak mixing (Halmos-von Neumann), and positive entropy can be fully realized by a Bernoulli-shift factor (Sinai). The second of these results was recently strengthened to show that all ergodic systems have Thouvenot's weak Pinsker property: they can always be split as a direct product of (i) a system with arbitrarily little entropy and (ii) a Bernoulli shift.

I will give a rough overview of some recent developments in this area and of some related settings in which many questions remains open. For the latter, I will especially emphasize the world of measure-preserving actions of sofic, non-amenable groups such as free groups.

Oscar Bandtlow, Queen Mary University of London, UK

EXPLICIT RESONANCES FOR ANALYTIC HYPERBOLIC MAPS

Joint work with Wolfram Just and Julia Slipantschuk

Date: 2019-09-16 (Monday); Time: 11:15-11:35; Location: building A-4, room 106.

Abstract

In a seminal paper Ruelle showed that the long-time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of the eigenvalues, also known as Pollicott-Ruelle resonances, of the so-called Ruelle transfer operator, a compact operator acting on a suitable Banach space of holomorphic functions.

In this talk I will report on recent work with Wolfram Just and Julia Slipantschuk on how to construct analytic expanding circle maps or analytic Anosov diffeomorphisms on the torus with explicitly computable non-trivial Pollicott-Ruelle resonances. I will also discuss applications of these results.

Zoltán Buczolich, Eötvös Loránd University, Hungary

GENERIC BIRKHOFF SPECTRA

Joint work with Balázs Maga and Ryo Moore

Date: 2019-09-17 (Tuesday); Time: 16:20-16:40; Location: building A-4, room 106.

Abstract

Suppose that \(\Omega = \{0, 1\}^ {\mathbb {N}}\) and \(\sigma\) is the one-sided shift. The Birkhoff spectrum \( \displaystyle S_{f}( {\alpha})=\dim_{H}\Big \{ {\omega}\in {\Omega}:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(\sigma^n \omega) = \alpha \Big \},\) where \(\dim_{H}\) is the Hausdorff dimension. It is well-known that the support of \(S_{f}( {\alpha})\) is a bounded and closed interval \(L_f = [\alpha_{f, \min}^*, \alpha_{f, \max}^*]\) and \(S_{f}( {\alpha})\) on \(L_{f}\) is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical \(f\in C( \Omega)\) in the sense of Baire category. For a dense set in \(C( \Omega)\) the spectrum is not continuous on \(\mathbb {R}\), though for the generic \(f\in C(\Omega)\) the spectrum is continuous on \( \mathbb {R}\), but has infinite one-sided derivatives at the endpoints of \(L_{f}\). We give an example of a function which has continuous \(S_{f}\) on \(\mathbb {R}\), but with finite one-sided derivatives at the endpoints of \(L_{f}\). The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions \(f\) and \(g\) are close in \(C(\Omega)\) then \(S_{f}\) and \(S_{g}\) are close on \(L_{f}\) apart from neighborhoods of the endpoints.

David Burguet, CNRS & Sorbonne Université, France

SYMBOLIC EXTENSIONS AND UNIFORM GENERATORS FOR TOPOLOGICAL REGULAR FLOWS

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building A-4, room 106.

Abstract

Building on the theory of symbolic extensions and uniform generators for discrete transformations we develop a similar theory for topological regular flows. In this context a symbolic extension is given by a suspension flow over a subshift.

References

  1. D. Burguet, Symbolic extensions and uniform generators for topological regular flows, Journal of Differential Equations, (to appear), https://arxiv.org/abs/1812.04285.

Nishant Chandgotia, Hebrew University of Jerusalem, Israel

SOME RESULTS ON PREDICTIVE SEQUENCES

Joint work with Benjamin Weiss

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building A-4, room 106.

Abstract

A sequence of natural numbers \(P\) is called predictive if for any zero-entropy stationary process \(X_i, X_0\) is measurable with respect to \(X_{-i}\); \(i\in P\). In this talk, we will discuss several necessary conditions and sufficient conditions for sequences to be predictive.

Alexandre Danilenko, Institute for Low Temperature Physics and Engineering, NAS, Ukraine

GENERIC NONSINGULAR POISSON SUSPENSION IS OF TYPE \(III_1\)

Joint work with Emmanuel Roy and Zemer Kosloff

Date: 2019-09-17 (Tuesday); Time: 16:45-17:05; Location: building A-4, room 106.

Abstract

Let \((X,\mu)\) be a standard measure space equipped with a non-atomic \(\sigma\)-finite infinite measure and let Aut\((X,\mu)\) denote the group of all \(\mu\)-nonsingular transformations of \(X\). The Poisson suspension \((X^*,\mu^*)\) of \((X,\mu)\) is a well defined Lebesgue space. Then \[ \text{Aut}_2(X,\mu):= \left\{T\in \text{Aut}(X,\mu)\mid \sqrt{\frac{d\mu\circ T}{d\mu}}-1\in L^2(X,\mu)\right\} \] is the largest subgroup of Aut\((X,\mu)\) consisting of those \(T\) for which the Poisson suspension \(T_*\) is \(\mu^*\)-nonsingular [1]. It contains strictly the group \[ \text{Aut}_1(X,\mu):= \left\{T\in \text{Aut}(X,\mu)\mid {\frac{d\mu\circ T}{d\mu}}-1\in L^1(X,\mu)\right\} \] introduced in [2]. Aut\(_j(X,\mu)\) admits a natural Polish topology \(d_j\) stronger than the weak topology, \(j=1,2\), and \(d_1\) is stronger then \(d_2\) [1]. There is a continuous homomorphism \(\chi:\operatorname{Aut}_1(X,\mu)\to \Bbb R\), \(\chi(T):=\int_X(\frac{d\mu\circ T}{d\mu}-1)d\mu\) [1, 2].

Theorem 1. \(\{T\in \operatorname{Aut}_2(X,\mu)\mid T\text{ is ergodic of type \(III_1\) and \(T_*\) is ergodic of type \(III_1\)}\}\) is a dense \(G_\delta\) in \(d_2\).

Theorem 2. \(\{T\in \text{Ker} \chi\mid T\text{ is ergodic of type \(III_1\) and \(T_*\) is ergodic of type $III_1$}\}\) is a dense \(G_\delta\) in \((\text{Ker }\chi,d_1)\).

Theorem 3. If \(T\in\operatorname{Aut}_1(X,\mu)\) and \(\chi(T)\ne 0\) then \(T_*\) is totally dissipative.

Example. There is a totally dissipative \(T\in \operatorname{Aut}_1(X,\mu)\) such that for each real \(t\in (0,\frac 1{4})\), the Poisson suspension \((X^*,(t\mu)^*,T_*)\) is conservative but for each \(t>2\), the Poisson suspension \((X^*,(t\mu)^*,T_*)\) is totally dissipative.

References

  1. A.I. Danilenko, E. Roy, Z. Kosloff, Nonsingular Poisson suspensions, in preparation.
  2. Yu A. Neretin, Categories of symmetries and infinite-dimensional groups, Oxford University Press, 1996.

Thierry de la Rue, CNRS & Université de Rouen Normandie, France

SARNAK CONJECTURE IN DENSITY

Joint work with Alexander Gomilko and Mariusz Lemańczyk

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-4, room 106.

Abstract

The talk will be based on a recent joint work with Alexander Gomilko and Mariusz Lemańczyk, in which we establish the following result related to Sarnak conjecture. (Here \(\mu\) denotes the classical Möbius arithmetic function.)

If \((X,T)\) is a zero entropy topological dynamical system with at most countably many invariant measures, then there exists a subset \(A=A(X,T)\) of full logarithmic density in the set of natural integers, such that for any \(f\) continuous on \(X\), \[ \sup_x \frac{1}{N} \sum_{1\le n\le N} \mu(n) f(T^n x) \longrightarrow 0, \quad\text{as }N\to\infty,\ N\in A. \]

The main tools are the results about logarithmic Furstenberg systems of the Möbius function proved by Frantzikinakis and Host [1], the logarithmic version of the so-called strong MOMO property [2], and an argument inspired by Tao to pass from logarithmic averages to classical averages along a subsequence of full logarithmic density.

References

  1. N. Frantzikinakis, B. Host, The logarithmic Sarnak conjecture for ergodic weights, Annals Math. 187 (2018), 869–931.
  2. E. H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk, T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math. 228 (2018), 707–751.

Tomasz Downarowicz, Wrocław University of Science and Technology, Poland

A CRUSH COURSE ON SYMBOLIC EXTENSIONS OF \(\mathbb{Z}\)-ACTIONS

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

Given a dynamical system \((X,T)\), where \(T\) is a homeomorphism of a compact metric space \(X\), we seek for its lossless digitalization in form of a subshift \((Y,\sigma)\), where \(Y\subset\Lambda^\mathbb Z\) (\(\Lambda\) is a finite alphabet) and \(\sigma\) denotes the standard shift, such that \((X,T)\) is a topological factor of \((Y,\sigma)\). It is obvious that a symbolic extension exists only for systems with finite topological entropy. But this condition is not sufficient. It turns out that in order to decide which systems admit symbolic extensions and how small can be their entropy one needs to study subtle entropy properties captured by the so-called entropy structure. In my talk I will try to present the most crucial definitions and facts around this topic.

References

  1. M. Boyle, T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math. 156 (2004), 119-161.
  2. T. Downarowicz, Entropy in Dynamical Systems, Cambridge University Press, Cambridge, 2011.

Aurelia Dymek, Nicolaus Copernicus University in Toruń, Poland

\(\mathcal{B}\)-FREE NUMBERS FROM DYNAMICAL POINT OF VIEW

Joint work with Stanisław Kasjan, Joanna Kułaga-Przymus and Mariusz Lemańczyk

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building A-4, room 106.

Abstract

Let \(\mathcal{B}\subset\{2,3,\ldots\}\). We call an integer \(n\) a \(\mathcal{B}\)-free number if \(n\) has no factor in \(\mathcal{B}\). We denote the set of all \(\mathcal{B}\)-free integers by \(\mathcal{F}_{\mathcal{B}}\). We consider the characteristic function of \(\mathcal{F}_\mathcal{B}\) in the space of binary sequences and denote it by \(\eta\). The subshift given by the orbit closure of \(\eta\) is called \(\mathcal{B}\)-free system and denoted by \(X_\eta\). A prominent example of a such system is the square-free system which is studied since 2010 [2]. In this case the frequencies of blocks yields a natural shift-invariant ergodic measure on \(\{0,1\}^\mathbb{Z}\). It is called the Mirsky measure.

During the talk I will concentrate on same ergodic properties of \(\mathcal{B}\)-free systems (genericity, entropy, invariant measures) and give some combinatorial applications [1].

References

  1. A. Dymek, S. Kasjan, J. Kułaga-Przymus, M. Lemańczyk, \(\mathcal{B}\)-free sets and dynamics, Trans. Amer. Math. Soc. 370(8) (2018), 5425–5489.
  2. P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.

Peyman Eslami, University of Rome Tor Vergata, Italy

INDUCING SCHEMES FOR PIECEWISE EXPANDING MAPS OF \(\mathbb{R}^{n}\)

Date: 2019-09-16 (Monday); Time: 11:40-12:00; Location: building A-4, room 106.

Abstract

For piecewise expanding maps of \(\mathbb{R}^{n}\) I will show how to construct an inducing scheme where the base map is Gibbs-Markov and the return times have exponential tails. The existence of such a structure has many consequences in regards to the statistical properties of systems with discontinuities and non-uniform expansion.

Matthew D. Foreman, University of California, Irvine, USA

PROGRESS ON THE SMOOTH REALIZATION PROBLEM

Joint work with Benjamin Weiss

Date: 2019-09-19 (Thursday); Time: 11:50-12:10; Location: building A-4, room 106.

Abstract

We discuss a Global Structure Theorem for measure preserving transformations that has two corollaries:

1. For all Choquet simplices \(\mathcal K\) there is an ergodic Lebesgue-measure preserving diffeomorphism of the 2-torus that has \(\mathcal K\) as its simplex of invariant measures.

2. For all countable ordinals \(\alpha\) there is a measure distal, measure preserving diffeomorphism of the 2-torus that has distal height \(\alpha\).

The first result changes Toeplitz systems built by Downarowicz into transformations that can be realized as diffeomorphisms. The second result stands in contrast to work of Rees, who showed that in the category of topological distality, the distal height is bounded by the dimension of the manifold.

References

  1. J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. 33(2) (1932), 587-642.
  2. D.V. Anosov, A.B. Katok, New examples in smooth ergodic theory, Trudy Moskov. Mat. Obšč. 23 (1970), 3-36.
  3. T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math. 74 (1991), 241-256.
  4. M. Rees, On the structure of minimal distal transformation groups with topological manifolds as phase spaces, Ph.D. Thesis, University of Warwick (1977).
  5. M. Foreman and B. Weiss, A Symbolic Representation of Anosov-Katok systems, Journal d’Analyse Mathématique 137 (2019), 603-661.
  6. M. Foreman and B. Weiss, From Odometers to Circular Systems: A Global Structure Theorem, ArXiv 1703.07093, March (2017).

Thomas M. Jordan, University of Bristol, UK

MULTIFRACTAL ANALYSIS FOR PLANAR SELF-AFFINE SETS

Joint work with Balázs Bárány, Antti Käenmäki, and Michał Rams

Date: 2019-09-17 (Tuesday); Time: 17:10-17:30; Location: building A-4, room 106.

Abstract

There is a standard problem in multifractal analysis of looking at level sets determined by the Birkhoff average of a suitable function. We look at the problem for self-affine sets on the plane. We show how recent work by Bárány, Hochman and Rapaport combined with results on approximation of pressure functions on suitable subsystems can give fairly complete solutions to this problem under certain generic algebraic assumptions and suitable separation assumptions.

Olena Karpel, AGH University of Science and Technology, Poland & Institute for Low Temperature Physics and Engineering, NAS, Ukraine

THE NUMBER OF ERGODIC INVARIANT MEASURES FOR BRATTELI DIAGRAMS

Joint work with Sergey Bezuglyi and Jan Kwiatkowski

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building A-4, room 106.

Abstract

We study the simplex \(\mathcal{M}_1(B)\) of probability measures on a Bratteli diagram \(B\) which are invariant with respect to the tail equivalence relation. Equivalently, \(\mathcal{M}_1(B)\) is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We prove a criterion of unique ergodicity of a Bratteli diagram. In the case of a finite rank \(k\) Bratteli diagram \(B\), we give a criterion for \(B\) to have exactly \(1 \leq l \leq k\) ergodic invariant measures and describe the structures of the diagram and the subdiagrams which support these measures. We also find sufficient conditions under which a Bratteli diagram of arbitrary rank has a prescribed number (finite or infinite) of probability ergodic invariant measures.

Gerhard Keller, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

PERIODS AND FACTORS OF WEAK MODEL SETS

Joint work with Christoph Richard

Date: 2019-09-17 (Tuesday); Time: 17:35-17:55; Location: building A-4, room 106.

Abstract

Weak model sets are defined by a cut-and-project scheme (CPS) \((G,H,\mathcal{L},W)\), where \(G\) and \(H\) are locally compact second countable abelian groups, \(\mathcal{L}\subset G\times H\) is a cocompact lattice, and \(W\subset H\) is a compact set called the window. Denote by \(\pi_G\) and \(\pi_H\) the canonical projections from \(G\times H\) to \(G\) and \(H\), respectively. It is assumed that \(\pi_G|_{\mathcal{L}}\) is \(1-1\) and that \(\pi_H(\mathcal{L})\) is dense in \(H\). Typical cases to think of are \(G=\mathbb{Z}^d\) or \(\mathbb{R}^d\), while \(H\) could be \(\mathbb{R}^k\) or an odometer group.

Denote \(\mathbb{T}:=(G\times H)/\mathcal{L}\). For \(t\in\mathbb{T}\) the set \(\Lambda_t:=\pi_G\left((G\times W)\cap(t+\mathcal{L})\right)\subset G\) is a weak model set. The structure of \(\Lambda_t\) can be studied (besides many other possibilities) using dynamical systems methods: To that end define \(X:=\overline{\{\Lambda_t:t\in\mathbb{T}\}}\) and its subset \(X_0:=\overline{\{\Lambda_0+g:g\in G\}}\), where the topology stems from the vague topology on the space of locally finite measures on \(G\) when \(\Lambda\subset G\) is identified with the Dirac comb \(\sum_{x\in\Lambda}\delta_x\). In this way both spaces are compact metrizable, and \(G\) acts on them by translation.

Model sets, i.e. the case when \(\overline{\operatorname{int}(W)}=W\), were originally studied by Y. Meyer [4], motivated by problems in harmonic analysis. There dynamical aspects are much studied and well understood. If \(W\) is aperiodic (\(W+h=W\) \(\Rightarrow\) \(h=0\), always true if \(H=\mathbb{R}^d\)), then \((X,G)\) is an almost 1-1 extension of its maximal equicontinuous factor (MEF) \((\mathbb{T},G)\), and if also \(|\partial W|=0\), then Haar-a.e. fibre of this factor map is a singleton. Examples are Sturmian sequences, Toeplitz sequences, the set of vertices of a Penrose tiling, and many others. See [5, 1] for reference.

But also the situation when \(\overline{\operatorname{int}(W)}\) is a strict subset of \(W\) is of considerable interest; it suffices to mention the set of square free integers or the visible lattice points, which are weak model sets with compact groups \(H\) and \(\operatorname{int}(W)=\emptyset\). Sets of \(\mathcal{B}\)-free numbers provide many other, intermediate examples. In [2] we prove among others:

Theorem A. \((\mathbb{T}/_{\mathbb{H}_{\operatorname{int}(W)}},G)\) is the MEF of \((X,G)\), where \(\mathbb{H}_{\operatorname{int}(W)}=\{(0,h)\in G\times H: \operatorname{int}(W)+h=\operatorname{int}(W)\}\). If \(\mathbb{H}_W=\mathbb{H}_{\operatorname{int}(W)}\), this is an almost 1-1 extension.

Remark. If \(\operatorname{int}(W)=\emptyset\), the MEF is thus trivial. But if \(W\) is aperiodic and Haar regular, the maximal equicontinuous generic factor is still \((\mathbb{T},G)\), see [3].

Theorem B. \((X,G,Q)\) is measure theoretically isomorphic to \((\mathbb{T}/_{{\mathbb{H}}_W^{Haar}},G,|\,.\,|)\), where \(|\,.\,|\) denotes Haar measure, \({{\mathbb{H}}_W^{Haar}}=\{(0,h)\in G\times H: |(W+h)\triangle W|=0\}\), and \(Q\) is the image of the Haar measure on \(\mathbb{T}\) under the map \(t\mapsto \Lambda_t\) (called Mirsky measure in arithmetic contexts).

References

  1. M. Baake, D. Lenz, R.V. Moody, Characterization of model sets by dynamical systems, Ergod. Th. & Dynam. Sys. 27 (2007), 341-382.
  2. G. Keller, C. Richard, Periods and factors of weak model sets, Israel J. Math. 229 (2019), 85-132.
  3. G. Keller, Maximal equicontinuous generic factors and weak model sets, ArXiv:1610.03998 (2016).
  4. Y. Meyer, Algebraic Numbers and Harmonic Analysis, North-Holland, Amsterdam, 1972.
  5. M. Schlottmann, Generalized model sets and dynamical systems in: Directions in Mathematical Quasicrystals, Eds. M.~Baake, R.V.~Moody, CMR Monograph Series 13 (2000), 143-159.

Jakub Konieczny, Hebrew University of Jerusalem, Israel

AUTOMATIC SEQUENCES, NILSYSTEMS, AND HIGHER ORDER FOURIER ANALYSIS

Joint work with Jakub Byszewski and Clemens Müllner

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building A-4, room 106.

Abstract

Automatic sequences give rise to one of the basic models of computation and have remarkable links to many areas of mathematics, including dynamics, algebra and logic. Distribution of these sequences has long been studied. During the talk we will explore this topic from the point of view of higher order Fourier analysis. As it turns out, many of the classical automatic sequences are highly Gowers uniform, while others can be expressed as the sum of a structured component and a uniform component much more efficiently than guaranteed by the arithmetic regularity lemma. We investigate the extent to which this phenomenon extends to general automatic sequences and consider some closely related problems that make sense for sparse sequences. 

Mariusz Lemańczyk, Nicolaus Copernicus University in Toruń, Poland

MULTIPLICATIVE FUNCTIONS AND DISJOINTNESS IN ERGODIC THEORY

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

In 2010, P. Sarnak formulated the Möbius orthogonality conjecture stating that the classical Möbius function does not correlate with any continuous observable in a (topological) zero entropy dynamical system. This conjecture has deep connections with analytic number theory and joinings in ergodic theory. My talk will be devoted to present some of these connections and an overview of the latest achievements.

Jacek Serafin, Wrocław University of Science and Technology, Poland

A STRICTLY ERGODIC, POSITIVE ENTROPY SUBSHIFT UNIFORMLY UNCORRELATED TO THE MÖBIUS FUNCTION

Joint work with Tomasz Downarowicz

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-4, room 106.

Abstract

This talk is based on two recent papers [1] and [2], where we show that if \(y=(y_n)_{n\ge 1}\) is a bounded sequence with zero average along every infinite arithmetic progression then for every \(N\ge 2\) there exists a strictly ergodic subshift \(\Sigma\) over \(N\) symbols, with entropy arbitrarily close to \(\log N\), uniformly uncorrelated to \(y\). In particular, for \(y=\mu\) being the Möbius function, there exist subshifts as above which satisfy the assertion of Sarnak’s conjecture ([3]). To the best of our knowledge, no other examples of positive entropy systems uncorrelated to the Möbius sequence are known.

Our result shows, among other things, that (even for strictly ergodic systems) the so-called strong MOMO (Möbius Orthogonality on Moving Orbits) property is essentially stronger than uniform uncorrelation.

References

  1. T. Downarowicz, J. Serafin, Almost full entropy subshifts uncorrelated to the Möbius function, International Mathematics Research Notices (2017), https://doi.org/10.1093/imrn/rnx192.
  2. T. Downarowicz, J. Serafin, A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Möbius function, Studia Mathematica (2019), to appear.
  3. P. Sarnak, Three lectures on the Möbius function randomness and dynamics, http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.

Tomasz Szarek, University of Gdańsk, Poland

INVARIANT MEASURES FOR RANDOM WALKS ON HOMEO\(^+\)(R)

Joint work with D. Buraczewski and S. Brofferio

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building A-4, room 106.

Abstract

Let \({g_n}\) be a sequence of i.i.d. Homeo\(^+\)(R)–valued randomvariables whose distribution is a measure \(\mu\). We consider the left randomwalk on Homeo\(^+\)(R) defined by the random variables \(f_n := g_n \circ\cdot\cdot\cdot\circ g_1\). We study the Markov chain \((X_n)\) on the real line corresponding to \({g_n}\), i.e.for any \(x \in \mathbb{R}\) and \(n \in \mathbb{N}\) we consider the random variables defined by \(X^x_n :=f_n(x)\). The main purpose of the talk is to provide suffcient conditions forthe existence of a unique invariant Radon measure (mainly infinite) for\((X_n)\). This research generalizes the results obtained by Deroin, Kleptsyn,Navas and Parvani, who studied similar problems for groups of homeomorphisms.

Dalia Terhesiu, Leiden University, Netherlands

LIMIT PROPERTIES FOR WOBBLY INTERMITTENT MAPS

Joint work with Douglas Coates and Mark Holland

Date: 2019-09-19 (Thursday); Time: 17:35-17:55; Location: building A-4, room 106.

Abstract

It is known that finite measure preserving intermittent maps with indifferent fixed points characterised by regular variation satisfy stable laws for sufficiently regular observables that do not vanish at the indifferent fixed points. We consider a finite measure preserving Pomeau Manneville type map, perturb the behaviour at the (only one) indifferent fixed point according to a St. Petersburg type distribution and obtain convergence to a non trivial limit distribution (a semistable law) along subsequences. Also, we obtain lower bounds on the decay of correlation for such modified maps and suitable observables. In this talk I will present these results.

Masato Tsujii, Kyushu University, Japan

COHOMOLOGICAL THEORY OF THE SEMI-CLASSICAL ZETA FUNCTIONS

Date: 2019-09-16 (Monday); Time: 10:40-11:10; Location: building A-4, room 106.

Abstract

We first review very briefly about recent developments in analysis of transfer operators for hyperbolic dynamical systems. We will then focus on the semi-classical (or Gutzwiller-Voros) zeta functions for geodesic flows on negatively curved manifolds. We show that the semi-classical zeta function is the dynamical Fredholm determinant of a transfer operator acting on the leaf-wise cohomology space along the unstable foliation. This realize the idea presented by Guillemin and Patterson a few decades ago. As an application, we see that the zeros of the semi-classical zeta function concentrate along the imaginary axis, imitating those of Selberg zeta function.

References

  1. F. Faure, M. Tsujii, The semiclassical zeta function for geodesic flows on negatively curved manifolds, Inventiones Mathematicae 208 (2017), 851-998.
  2. M. Tsujii, On cohomological theory of dynamical zeta functions, preprint, arXiv 1805.11992.

Thomas Ward, University of Leeds, UK

TIME-CHANGES PRESERVING ZETA FUNCTIONS

Joint work with Sawian Jaidee and Patrick Moss

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building A-4, room 106.

Abstract

A time-change is a function \(h\colon\mathbb{N}\to\mathbb{N} \), and \(h\) is said to 'preserve zeta functions' if, for any dynamical zeta function \(\exp\left(\sum_{n\ge1}a_nz^n/n\right)\), where \(a_n=\vert\{x\in X\mid T^nx=x\}\vert\) for some dynamical system \(T\colon X\to X\), the time-changed function \(\exp\left(\sum_{n\ge1}a_{h(n)}z^n/n\right)\) is the dynamical zeta function of some dynamical system. That is, for any homeomorphism of a compact metric space \(T\colon X\to X\) there is some other homeomorphism of a compact metric space \(S\colon Y\to Y\) with the property that \(\vert\{x\in X\mid T^{h(n)}x=x\}\vert = \vert\{y\in Y\mid S^ny=y\}\vert\) for all \(n\in\mathbb{N}\). The time-changes that preserve zeta functions form a monoid \(\mathcal{P}\), and we show that a polynomial lies in \(\mathcal{P}\) if and only if it is a monomial (meaning that \(\mathcal{P}\) is algebraically small), that \(\mathcal{P}\) is uncountable (meaning that it is set-theoretically large), and that \(\mathcal{P}\) contains no permutations (that is, \(\mathcal{P}\) has no torsion as a monoid).

References

  1. S. Jaidee, P. Moss, T. Ward, Time-changes preserving zeta functions, Proc. Amer. Math. Soc. (to appear).
  2. A. Pakapongpun, T. Ward, Functorial orbit counting, J. Integer Seq. 12(2) (2009), Article 09.2.4.
  3. Y. Puri, T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seq. 4(2) (2001), Article 01.2.1.
  4. A. J. Windsor, Smoothness is not an obstruction to realizability, Ergodic Theory Dynam. Systems 28(3) (2008), 1037-1041.

Benjamin Weiss, Hebrew University of Jerusalem, Israel

ON THE COMPLEXITY OF SMOOTH SYSTEMS

Joint work with Matthew Foreman

Date: 2019-09-19 (Thursday); Time: 11:15-11:45; Location: building A-4, room 106.

Abstract

About ten years ago, in joint work with the late Dan Rudolph and Matt Foreman, we showed that the isomorphism relation for ergodic measure systems is not Borel, but rather a complete analytic set. In fact we showed that the transformations that are isomorphic to their inverses is already complete analytic. Since the smooth realization problem is still open it was not clear how complex is the class of diffeomorphisms of compact manifolds that preserve a volume element. In more recent work with Matt Foreman we show that already the ergodic diffeomorphisms of the torus that preserve Lebesgue measure is also a complete analytic set.

Guohua Zhang, Fudan University, China

ASYMPTOTIC \(h\)-EXPANSIVENESS FOR AMENABLE GROUP ACTIONS

Joint work with Tomasz Downarowicz

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building A-4, room 106.

Abstract

Asymptotic \(h\)-expansiveness for amenable group actions can be introduced respectively using topological conditional entropy in [1] and using entropy structure in [2]. In this talk we will show the equivalence of these two kinds of asymptotic \(h\)-expansiveness.

References

  1. N.-P. Chung and G. Zhang, Weak expansiveness for actions of sofic groups, J. Funct. Anal. 268(11) (2015), 3534-3565.
  2. T. Downarowicz and G. Zhang, Symbolic extensions of amenable group actions and the comparison property, arXiv:1901.01457, preprint.

Abed Bounemoura, CNRS & CEREMADE, Université Paris-Dauphine, France

ANALYTIC INVARIANT CURVES FOR ANALYTIC TWIST MAPS

Date: 2019-09-20 (Friday); Time: 11:15-11:35; Location: building A-4, room 120.

Abstract

Yoccoz proved that the Bruno arithmetical condition is optimal for the analytic linearization of a circle diffeomorphism close to a rotation. We will explain how to use this result to show that the same condition is optimal for the analytic preservation of quasi-periodic invariant curves for twist maps of the annulus, as well as new questions in this context.

Leo Butler, University of Manitoba, Canada

INVARIANT TORI FOR A CLASS OF THERMOSTATED SYSTEMS

Date: 2019-09-19 (Thursday); Time: 16:20-16:40; Location: building A-4, room 120.

Abstract

A thermostated hamiltonian system is a model of a mechanical system immersed in a heat bath at constant temperature. A fundamental question is whether the system reaches thermal equilibrium. Even for 1-degree-of-freedom hamiltonians this question is non-trivial, and an answer has only been known in the simplest case by work of Legoll, Luskin and Moeckel [4, 5].

Most of the literature on thermostats focuses on specific examples. I propose a mathematical definition of a thermostat that captures the content of these examples. Under some conditions, the properties of a thermostat lead to the existence of a thermostatic equilibrium when it is only weakly coupled with the hamiltonian. Under a modest additional hypothesis on the non-degeneracy of this equilibrium, one obtains the existence of invariant tori in a neighbourhood of the thermostatic equilibrium. The existence of such KAM tori frustrates "thermalization".

These conditions are verified for four well-known examples in the literature [1, 2, 3, 7, 8], when the hamiltonian is real-analytic and "well-behaved".

If time permits, I will discuss related results about variants including non-equilibrium thermostats, and multiple/recursive thermostats.

References

  1. W. Hoover, Canonical dynamics: equilibrium phase space distributions, Phys. Rev. A. 31 (1985), 1695–1697.
  2. W. Hoover, Nosé-Hoover nonequilibrium dynamics and statistical mechanics, Mol. Simul. 33 (2007), 13–19.
  3. W.G. Hoover, J.C. Sprott, and C.G. Hoover, Ergodicity of a singly-thermostated harmonic oscillator, Communications in Nonlinear Science and Numerical Simulation, 32(Supplement C):234 – 240, 2016.
  4. F. Legoll, M. Luskin, and R. Moeckel, Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator, Arch. Ration. Mech. Anal. 184(3) (2007), 449–463.
  5. F. Legoll, M. Luskin, and R. Moeckel, Non-ergodicity of Nosé-Hoover dynamics, Nonlinearity 22(7) (2009), 1673–1694.
  6. S. Nosé, A unified formulation of the constant temperature molecular dynamics method, J. Chem. Phys. 81 (1984), 511–519.
  7. D. Tapias, A. Bravetti, and D.P. Sanders, Ergodicity of one-dimensional systems coupled to the logistic thermostat, CMST 23, 11 2016.
  8. H. Watanabe and H. Kobayashi, Ergodicity of a thermostat family of the Nosé-Hoover type, Phys. Rev. E 75:040102, Apr 2007.

Bernardo Carvalho, Federal University of Minas Gerais, Brazil & Friedrich-Schiller-Universität Jena, Germany

CONTINUUM-WISE HYPERBOLICITY

Joint work with Alfonso Artigue, Welington Cordeiro, and José Vieitez

Date: 2019-09-16 (Monday); Time: 11:05-11:25; Location: building A-4, room 120.

Abstract

Hyperbolicity is one of the most important concepts in the theory of chaotic dynamical systems. Since the seminal works of Anosov [1] and Smale [5] it has been a main topic of research among many mathematicians. In hyperbolic systems, each tangent space splits into two invariant subspaces, the first being uniformly contracted, and the second uniformly expanded, by the action of the derivative map. The dynamics of such systems can be well described in both topological and statistical viewpoints, so many effort is being made to understand the dynamics beyond uniform hyperbolicity and many generalizations have been considered.

In this talk I will discuss one specific notion of hyperbolicity introduced in a joint work with A. Artigue, W. Cordeiro and J. Vieitez [3] called continuum-wise hyperbolicity. Examples of these systems are the Anosov diffeomorphisms [1], the topologically hyperbolic homeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere. We discuss the dynamics of cw-hyperbolic homeomorphisms, proving that some dynamical properties that are present in the hyperbolic theory, such as the shadowing property and a spectral decomposition, are still present in cw-hyperbolic ones, while enlightening the differences and peculiarities of these systems, such as wilder local stable sets containing cantor sets and the existence of arbitrarily small horseshoes.

References

  1. D.V. Anosov, Geodesic flows on compact manifolds of negative curvature, Trudy mat. Inst. V.A Steklova, 90 (1967).
  2. A. Artigue, B. Carvalho, W. Cordeiro, J. Vieitez, Beyond topological hyperbolicity: the L-shadowing property, arXiv:1902.07578.
  3. A. Artigue, B. Carvalho, W. Cordeiro, J. Vieitez, Continuum-wise hyperbolicity, preprint.
  4. A. Artigue, B. Carvalho, W. Cordeiro, J. Vieitez, Countably and entropy expansive homeomorphisms with the shadowing property, preprint.
  5. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.

Danijela Damjanović, Royal Institute of Technology (KTH), Sweden

CENTRALIZER CLASSIFICATION FOR SOME PARTIALLY HYPERBOLIC MAPS

Joint work with Amie Wilkinson and Disheng Xu

Date: 2019-09-17 (Tuesday); Time: 10:40-11:10; Location: building A-4, room 120.

Abstract

In this talk I will survey recent advances on classification of centralizer for some conservative partially hyperbolic diffeomorphisms with one dimensional center foliation. I will describe how disintegration of volume along the center foliation [4] together with classification of higher rank partially hyperbolic abelian actions [1], [2] lead to classification results for centralizers [3]. I will also mention several conjectures concerning classification of centralizers.

References

  1. D. Damjanović and D. Xu, On conservative partially hyperbolic abelian actions with compact center foliation, arXiv 1706.03626.
  2. D. Damjanović, A. Wilkinson, D. Xu, Global rigidity of conservative partially hyperbolic actions with compact center foliation, in preparation.
  3. D. Damjanović, A. Wilkinson, D. Xu, Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms, arXiv:1902.05201.
  4. A. Avila, M. Viana, A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity II, in preparation.

Bassam Fayad, CNRS & Université Paris Diderot, France

INFINITE LEBESGUE SPECTRUM FOR SURFACE FLOWS

Joint work with Giovanni Forni and Adam Kanigowski

Date: 2019-09-20 (Friday); Time: 10:40-11:10; Location: building A-4, room 120.

Abstract

We study the spectral measures of conservative mixing flows on the two torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients : 1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, 2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows, such as horocyclic flows on the unit tangent bundle \(M\) of a compact hyperbolic surface.

References

  1. B. Fayad, G. Forni , A. Kanigowski, Lebesgue spectrum of countable multiplicity for conservative flows on the torus, arXiv 2019.

Todd Fisher, Brigham Young University, USA

EQUILIBRIUM STATES FOR CERTAIN PARTIALLY HYPERBOLIC ATTRACTORS

Joint work with Krerley Oliveira

Date: 2019-09-16 (Monday); Time: 16:20-16:40; Location: building A-4, room 120.

Abstract

I will discuss partially hyperbolic attractors introduced by Castro and Nascimento and show they have unique equilibrium states for natural classes of potentials. If the system is \(C^2\), then there is a unique equilibrium states for the geometric potential and its 1-parameter family. The proofs follow by applying general techniques developed by Climenhaga and Thompson. This is joint work with Krerley Oliveira.

Ana Cristina Moreira Freitas, University of Porto, Portugal

DYNAMICAL COUNTEREXAMPLES FOR THE USUAL INTERPRETATION OF THE EXTREMAL INDEX

Date: 2019-09-17 (Tuesday); Time: 16:55-17:15; Location: building A-4, room 120.

Abstract

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The existence of an Extremal Index less than 1 is associated to the occurrence of periodic phenomena, which is responsible for the appearance of clusters of exceedances. The Extremal Index usually coincides with the reciprocal of the mean of the limiting cluster size distribution. We build dynamically generated stochastic processes with an Extremal Index for which that relation does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at at least two points of the phase space, where one of them is an indifferent periodic point.

Jorge Milhazes Freitas, University of Porto, Portugal

RARE EVENTS FOR FRACTAL LANDSCAPES

Date: 2019-09-17 (Tuesday); Time: 17:20-17:40; Location: building A-4, room 120.

Abstract

We consider the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to fractal sets of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal geometrical structure.

José Pedro Gaivão, University of Lisbon, Portugal

BILLIARDS INSIDE STRICTLY CONVEX BODIES WITH POSITIVE TOPOLOGICAL ENTROPY

Joint work with M. Bessa, G. Del Magno, J.L. Dias, and M.J. Torres

Date: 2019-09-16 (Monday); Time: 17:35-17:55; Location: building A-4, room 120.

Abstract

In this talk we discuss the topological entropy of billiards inside strictly smooth convex bodies. We show that in a \(C^2\)-open and dense set of strictly convex bodies, the associated multidimensional billiard maps have positive topological entropy.

Alexey O. Kazakov, National Research University Higher School of Economics, Russia

WILD PSEUDOHYPERBOLIC ATTRACTOR IN A FOUR-DIMENSIONAL LORENZ MODEL

Joint work with Sergey Gonchenko and Dmitry Turaev

Date: 2019-09-19 (Thursday); Time: 11:15-11:35; Location: building A-4, room 120.

Abstract

We present an example of a new strange attractor. We show that it belongs to a class of wild pseudohyperbolic spiral attractors. A theory of pseudohyperbolic spiral attractors was proposed in [1], however examples of concrete systems of differential equations with such attractors were not known.

We consider the following system of differential equation \[ \left\{ \begin{array}{l} \dot x = \sigma (y - x), \\ \dot y = x (r-z) - y, \\ \dot z = xy - bz + \mu w, \\ \dot w = -b w - \mu z, \end{array} \right. \tag{1} \] where \(\sigma,r,b\) and \(\mu\) are parameters. This system can be viewed as a four-dimensional extension of the classical Lorenz model. We perform a series of numerical experiments with the strange attractor which exists in the system at \(\mu = 7, \sigma = 10, b = 8/3, r = 25\). We demonstrate that this attractor is indeed pseudohyperbolic and wild.

The pseudohyperbolicity [1, 2] is a key word here. It means that certain conditions hold which guarantee that every orbit in the attractor is unstable (i.e. it has a positive maximal Lyapunov exponent). Moreover, this instability property persists for all small perturbations of the system.

The wildness of the observed attractor means that it contains a "wild hyperbolic set" [3 ,4] - a uniformly hyperbolic invariant set which has a pair of orbits such that the unstable manifold of one orbit has a nontransversal intersection with the stable manifold of the other orbit in the pair and this property is preserved for all \(C^2\)-small perturbations.

Acknowledgments

This work was supported by RSF grant No. 17-11-01041.

References

  1. D. Turaev, L.P. Shilnikov, An example of a wild strange attractor, Sbornik: Mathematics 189(2) (1998), 291–314.
  2. D. Turaev, L.P. Shilnikov, Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors, Doklady Mathematics 77(1) (2008), 17–21.
  3. S.E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci. 50 (1979), 101–151.
  4. S.V. Gonchenko, D. Turaev, L.P. Shilnikov, On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case), Russian Acad. Sci. Dokl. Math. 47(2) (1993), 268–283.

Martin Leguil, University of Toronto, Canada

SPECTRAL DETERMINATION OF OPEN DISPERSING BILLIARDS

Joint work with Péter Bálint, Jacopo De Simoi, and Vadim Kaloshin

Date: 2019-09-16 (Monday); Time: 16:45-17:05; Location: building A-4, room 120.

Abstract

In an ongoing project with P. Bálint, J. De Simoi and V. Kaloshin, we have been studying the inverse problem for a class of open dispersing billiards obtained by removing from the plane a finite number of smooth strictly convex scatterers satisfying a non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift of finite type that provides a natural labeling of all periodic orbits. One direction we have investigated in [1] is whether it is possible to recover from the Marked Length Spectrum (i.e., the set of lengths of all periodic orbits together with their labeling) the local geometry near periodic points. In particular, we show in [1] that the Marked Length Spectrum determines the curvatures of the scatterers at the base points of \(2\)-periodic orbits, and the Lyapunov exponents of each periodic orbit. In a second work [2], we show that it is generically possible, in the analytic category and for billiard tables with two (partial) axial symmetries, to determine completely the geometry of those billiards from the purely dynamical data encoded in their Marked Length Spectrum.

References

  1. P. Bálint, J. De Simoi, V. Kaloshin, and M. Leguil, Marked Length Spectrum, homoclinic orbits and the geometry of open dispersing billiards, Communications in Mathematical Physics (2019), 1-45.
  2. J. De Simoi, V. Kaloshin, and M. Leguil, Marked Length Spectral determination of analytic chaotic billiards with axial symmetries, arXiv preprint arXiv:1905.00890 (2019).

Dongchen Li, Imperial College London, UK

PERSISTENT HETERODIMENSIONAL CYCLES IN PERIODIC PERTURBATIONS OF LORENZ-LIKE ATTRACTORS

Joint work with Dmitry Turaev

Date: 2019-09-19 (Thursday); Time: 11:40-12:00; Location: building A-4, room 120.

Abstract

We prove that heterodimensional cycles can be created by unfolding a pair of homoclinic tangencies in a certain class of \(C^{\infty}\) diffeomorphisms. This implies the existence of a \(C^2\)-open domain in the space of dynamical systems with a certain type of symmetry where systems with heterodimensional cycles are dense in \(C^{\infty}\). In particular, we describe a class of three-dimensional flows with a Lorenz-like attractor such that an arbitrarily small perturbation of any such flow can belong to this domain - in this case the corresponding heterodimensional cycles belong to a chain-transitive attractor of the perturbed flow.

Reza Mohammadpour, Polish Academy of Sciences, Poland

ON HAUSDORFF DIMENSION OF THIN NONLINEAR SOLENOIDS

Joint work with Michał Rams and Feliks Przytycki

Date: 2019-09-19 (Thursday); Time: 16:45-17:05; Location: building A-4, room 120.

Abstract

Let \(M=S^1\times \mathbb{D}\) be the solid torus, where \(\mathbb{D}=\{ v\in \mathbb{R}^{2} | |v|<1 \}\) carries the product distance \(d=d_{1}\times d_{2}\), and suppose \(f:M\rightarrow M\) such that \[ (x,y,z)\mapsto (\eta(x,y,z)\! \!\!\!\mod 2\pi , \lambda(x,y,z)+u(x) , \nu(x,y,z)+v(x)) \tag{1}\] is a smooth embedding map.

Bothe [1] was the first who obtained results on the dimension of the attractor of a thin linear solenoid where contraction rates are strong enough. Barriera, Pesin and Schemeling [2] established a dimension product structure of invariant measures in the course of proving the Eckmann Ruelle conjecture.

Conjecture. The fractal dimension of a hyperbolic set is (at least generically or under mild hypotheses) the sum of those of its stable and unstable slices, where fractal can mean either Hausdorff or upper box dimension.

In spite of the difficulties due to possible low regularity of the holonomies, indeed, Schmeling [4] found that solenoids often lack regular holonomies but the set of non-liptchitz points seemed to be rather small in the measure scene. Hasselblat and Schmeling [3] proved the conjecture for a class of thin linear solenoids. We prove the conjecture for a class of thin nonlinear solenoids of map (1).

References

  1. H. Bothe, The dimension of some solenoids, Ergodic Theory and Dynamical Systems, 15 (1995), 449-474.
  2. L. Barreira, Y. Pesin, J. Schmeling, Dimension and product structure of hyperbolic measures, Annals of Mathematics, 3 (1999), 755-783.
  3. B. Hasselblatt, J. Schmeling, Dimension product structure of hyperbolic sets, In Modern dynamical systems and applications, 331-345. Cambridge Univ. Press, Cambridge, 2004.
  4. J. Schmeling, Hölder continuity of the holonomy maps for hyperbolic basic sets, II. Math. Nachr. 170 (1994), 211-225.

Péter Nándori, Yeshiva University, USA

THE LOCAL LIMIT THEOREM FOR HYPERBOLIC DYNAMICAL SYSTEMS AND APPLICATIONS

Date: 2019-09-16 (Monday); Time: 17:10-17:30; Location: building A-4, room 120.

Abstract

We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that the MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Examples include reward renewal processes, Axiom A flows, as well as the systems admitting Young's tower, such as Sinai's billiard with finite horizon, suspensions over Pomeau-Manneville maps and geometric Lorenz attractors. Then we discuss two applications in infinite ergodic theory. First, we prove the mixing of global observables by some infinite measure preserving hyperbolic systems that are well approximated by periodic systems (examples include billiards with small potential field and various ping pong models). Here, global observables are functions that are not integrable with respect to the infinite invariant measure, but have convergent average values over large boxes. Second, we discuss the Birkhoff theorem for such global observables in the simplest case: iid random walks. The talk is based on joint work with Dmitry Dolgopyat and in parts with Marco Lenci.

References

  1. D. Dolgopyat, M. Lenci, P. Nándori, Global observables for random walks: law of large numbers, http://arxiv.org/abs/1902.11071 (2019).
  2. D. Dolgopyat, P. Nándori, Infinite measure mixing for some mechanical systems, http://arxiv.org/abs/1812.01174 (2018).
  3. D. Dolgopyat, P. Nándori, On mixing and the local central limit theorem for hyperbolic flows, Ergodic Theory and Dynamical Systems, to appear, https://doi.org/10.1017/etds.2018.29.

Ivan Ovsyannikov, Universität Hamburg, Germany

BIRTH OF DISCRETE LORENZ ATTRACTORS IN GLOBAL BIFURCATIONS

Date: 2019-09-19 (Thursday); Time: 12:05-12:25; Location: building A-4, room 120.

Abstract

Discrete Lorenz attractors are chaotic attractors, which are the discrete-time analogues of the well-known continuous-time Lorenz attractors. They are genuine strange attractors, i.e. they do not contain simpler regular attractors such as stable equilibria, periodic orbits etc. In addition, this property is preserved under small perturbations. Thus, Lorenz attractors, discrete and continuous, represent the so-called robust chaos.

In the talk a list of global (homoclinic and heteroclinic) bifurcations [1, 2, 3, 4] is presented, in which it was possible to prove the appearance of discrete Lorenz attractors. The proof is based on the study of first return (Poincare) maps, which are defined in a small neighbourhood of the homoclinic or heteroclinic cycle. The first return map can be transformed to the form asymptotically close to the three-dimensional Hénon map via smooth transformations of coordinates and parameters. According to [1, 5, 6, 7], Henon-like maps possess the discrete Lorenz attractor in an open subset of the parameter space.

References

  1. S. Gonchenko, J. Meiss, I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn. 11 (2006), 191–212.
  2. S. Gonchenko, I. Ovsyannikov, On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors, Math. Model. Nat. Phenom. 8 (2013), 71–83.
  3. S. Gonchenko, I. Ovsyannikov, J. Tatjer, Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points, Regul. Chaotic Dyn. 19 (2014), 495–505.
  4. S. Gonchenko, I. Ovsyannikov, Homoclinic tangencies to resonant saddles and discrete Lorenz attractors, Discrete Contin. Dyn. Syst. S 10 (2017), 273–288.
  5. S. Gonchenko, I. Ovsyannikov, C. Simo, D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), 3493–3508.
  6. S. Gonchenko, A. Gonchenko, I. Ovsyannikov, D. Turaev, Examples of Lorenz-like attractors in Hénon-like maps, Math. Model. Nat. Phenom. 8 (2013), 32–54.
  7. I. Ovsyannikov, D. Turaev, Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model, Nonlinearity 30 (2017), 115–137.

Bernardo San Martín, Catholic University of the North, Chile

THE ROVELLA ATTRACTOR IS ASYMPTOTICALLY SECTIONAL-HYPERBOLIC

Joint work with Kendry Vivas

Date: 2019-09-16 (Monday); Time: 10:40-11:00; Location: building A-4, room 120.

Abstract

The Rovella attractor is a compact invariant set for a vector field \(X_0\) constructed in a similar way as the geometric Lorenz attractor, but replacing the central expansive condition at the singularity by a central contracting condition plus two additional geometric hypothesis: the unstable manifold of the singularity is contained in the stable manifold of hyperbolic periodic orbits and the one dimensional reduction for the first return Poincaré map has negative Schwarzian derivative. Rovella showed that although this attractor is non robust, it is almost 2-persistent in the \(C^3\) topology. In this paper we will prove that for a generic two-parameter family of vector fields that contains \(X_0\), asymptotically sectional-hyperbolicity is an almost 2-persistent property. In particular, we will prove that the Rovella attractor is asymptotically sectional-hyperbolic.

References

  1. D. Carrasco-Olivera, B. San Martín, On the \(\mathcal{K}^*-\)expansiveness of the Rovella attractor, Bull. Braz. Math. Soc., 48 (2017), 649—662.
  2. R.J. Metzger and C.A. Morales, The Rovella attractor is a homoclinic class, Bull. Braz. Math. Soc., 37(206) (2006), 89—101.
  3. C.A. Morales and B. San Martín, Contracting Singular Horseshoe, Nonlinearity, 30 (2017), 4208–4219.
  4. E.M. Muñoz Morales, B. San Martín and J.A. Vera Valenzuela, Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. Inst. H. Poincaré Anal. Non Lineaire, 45–67, Series, 20 (2003), 867–888.
  5. A. Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Brasil. Mat, 24 (1993), 233-–259.
  6. B. San Martín and K. Vivas, Asymtoticaly sectional-hiperbolic attractors, Discrete Contin. Dyn. Syst., 39 (2019), 4057-4071.

Maria Saprykina, Royal Institute of Technology (KTH), Sweden

INSTABILITY AND DIFFUSION IN HAMILTONIAN SYSTEMS VIA THE APPROXIMATION BY CONJUGATION METHOD

Joint work with Bassam Fayad

Date: 2019-09-20 (Friday); Time: 11:40-12:00; Location: building A-4, room 120.

Abstract

We present examples of nearly integrable Hamiltonian systems with several strong diffusion properties. In particular, we construct a real-analytic near integrable Hamiltonian system whose flow is topologically weakly mixing on the energy surface.

Our constructions are obtained by a version of the successive conjugation scheme à la Anosov-Katok. The talk is based on a joint work with Bassam Fayad.

Dmitry Turaev, Imperial College London, UK

LORENZ ATTRACTORS IN FLOWS AND MAPS

Date: 2019-09-19 (Thursday); Time: 10:40-11:10; Location: building A-4, room 120.

Abstract

We review a theory of pseudohyperbolic attractors, which serve as a generalization of the classical Lorenz attractor to the case of a higher codimension of the strong-stable foliation. These attractors are genuinely chaotic (every orbit in such attractor has positive maximal Lyapunov exponent) and are robust with respect to small perturbations. The class includes periodically perturbed hyperbolic and Lorenz attractors and attractors in lattice dynamical systems. We show that pseudohyperbolic attractors emerge naturally at local and global bifurcations of codimension 3, hence they are present in a vast set of diverse applications. We also show that robust presence of homoclinic tangencies and heterodimensional cycles is a characteristic feature of pseudohyperbolic attractors.

Sebastian van Strien, Imperial College London, UK

CONJUGACY CLASSES OF REAL ANALYTIC MAPS: ON A QUESTION OF AVILA-LYUBICH-DE MELO

Joint work with Trevor Clark

Date: 2019-09-17 (Tuesday); Time: 16:20-16:50; Location: building A-4, room 120.

Abstract

Avila-Lyubich-de Melo proved that the topological conjugacy classes of unimodal real-analytic maps are complex analytic manifolds, which laminate a neighbourhood of any such mapping without a neutral cycle. Their proof that the manifolds are complex analytic depends on the fact that they have codimension-one in the space of unimodal mappings.

In joint work with Trevor Clark, we show how to construct a “pruned polynomial-like mapping" associated to a real mapping. This gives a new complex extension of a real-analytic mapping.

The additional structure provided by this extension, makes it possible to generalize this result of Avila-Lyubich-de Melo to interval mappings with several critical points. Thus we show that the conjugacy classes are complex analytic manifolds whose codimension is determined by the number of critical points.

Building on these ideas, we will show that in the space of unimodal mappings with negative Schwarzian derivative, the conjugacy classes laminate a neighbourhood of every mapping.

Kurt Vinhage, University of Chicago, USA

CLASSIFICATION OF TOTALLY CARTAN ACTIONS

Joint work with Ralf Spatzier

Date: 2019-09-17 (Tuesday); Time: 11:15-11:35; Location: building A-4, room 120.

Abstract

We will discuss recent progress on the Katok-Spatzier conjecture, which aims to classify Anosov \(\mathbb{R}^k\) and \(\mathbb{Z}^k\) actions under the assumption that there are no nontrivial smooth rank one factors. Classification is the strongest conclusion in the smooth rigidity program, which assumes nothing about the structure of the underlying manifold or dynamics other than the Anosov hyperbolicity assumptions. We develop new techniques to build homogeneous structures from dynamical ones. The remarkable features of the techniques are their low regularity requirements and their use of metric geometry over differential geometry to build group actions. We apply these techniques to the totally Cartan setting, where bundles associated to the hyperbolic strcuture are one-dimensional. Joint with Ralf Spatzier.

Daren Wei, Pennsylvania State University, USA

KAKUTANI EQUIVALENCE OF UNIPOTENT FLOWS

Joint work with Adam Kanigowski and Kurt Vinhage

Date: 2019-09-19 (Thursday); Time: 17:10-17:30; Location: building A-4, room 120.

Abstract

We study Kakutani equivalence in the class of unipotent flows acting on finite volume quotients of semisimple Lie groups. For every such flow we compute the Kakutani invariant of M. Ratner, the value of which being explicitly given by the Jordan block structure of the unipotent element generating the flow. This, in particular, answers a question of M. Ratner. Moreover, it follows that the only standard unipotent flows are given by \(\begin{pmatrix}1&t\\0&1\end{pmatrix}\times \operatorname{id}\) acting on \((\operatorname{SL}(2,\mathbb{R})\times G')/\Gamma'\), where \(\Gamma'\) is an irreducible lattice in \(\operatorname{SL}(2,\mathbb{R})\times G'\) (with the possibility that \(G' = \{e\}\)).

References

  1. A. Kanigowski, K. Vinhage, D. Wei, Kakutani equivalence of unipotent flows, preprint arXiv:1805.01501.

Disheng Xu, University of Chicago, USA

ON THE RIGIDITY OF PARTIALLY HYPERBOLIC \(\mathbb{Z}^k\) ACTION

Date: 2019-09-17 (Tuesday); Time: 11:40-12:00; Location: building A-4, room 120.

Abstract

Roughly speaking, a partially hyperbolic diffeomorphism on a manifold \(M\) is a certain type of mapping, from \(M\) to itself, with local directions of “expansion”, “neutral” and “contraction”. The study of the partially hyperbolic system, i.e. the \(\mathbb{Z}\)-action generated by the iterates of a partially hyperbolic diffeomorphism, is one of the central topic in dynamical systems in the last four decades.

On the other hand, in general it is expected that under suitable assumptions, a \(\mathbb{Z}^k\) action by diffeomorphisms on manifold, \(k > 1\) shares some strong rigidity properties (stronger than that of \(\mathbb{Z}\)-action). A \(\mathbb{Z}^k\)-action on a manifold is called partially hyperbolic if the action contains at least one partially hyperbolic diffeomorphism. In this talk we will show some recent rigidity results on the study of partially hyperbolic \(\mathbb{Z}^k\)-action on manifolds, this is a joint work with D. Damjanović and A. Wilkinson.

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