Zoominar in Dynamical Systems @ Porto

Date. May 19, 14h00 

Speaker   Pierre BERGER (Sorbonne Université)

Title.  Analytic pseudo-rotations

Abstract.  

 We construct analytic symplectomorphisms of the cylinder or the sphere with zero or exactly two periodic points and which are not conjugated to a rotation. In the case of the cylinder, we show that these symplectomorphisms can be chosen ergodic or to the contrary with local emergence of maximal order. In particular, this disproves a conjecture of Birkhoff (1941) and solve a problem of Herman (1998). One aspect of the proof provides a new approximation theorem, it enables in particular to implement the Anosov-Katok scheme in new analytic settings.

Date. May 05, 14h00 

Speaker   Dongkui MA (South China University)

Title.  The upper capacity topological entropy of free semigroup actions for certain non-compact sets

Abstract.  

In this talk, we introduce some new notions of `periodic-like' points, such as almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and g-almost product property of free semigroup actions. We find that the corresponding sets and gap-sets of these points carry full upper capacity topological entropy under certain conditions. Furthermore, $\phi$-irregular set and more general (ir)regular sets acting on free semigroup actions are introduced and they also carries full upper capacity topological entropy in the system with specification property or g-almost product property.   Finally, we introduce the level set for local recurrence of free semigroup actions and analyze its connections with upper capacity topological entropy. Our analysis generalizes the results obtained by Huang, Tian , Chen and Lau et al.

Date. April 21, 14h00 

Speaker   Martina MAIURIELLO (Universitá di Campania)

Title.  An exploration of linear, chaotic Koopman operators through lineability and spaceability

Abstract.  

Motivated by recent studies on the notions of lineability and spaceability in the context of linear dynamics, in this talk we shall investigate the “vastness” of chaotic phenomena in the context of linear operators, with a special focus on composition operators, also known in the literature as Koopman operators, and extensively used in many applications (like, for instance, the analysis of the dynamics of biological or economic models formulated in terms of dynamical systems). Chaos will be analyzed both, in the classical sense of Baire and in terms of lineability and spaceability. In this walk through the investigation of topological and linear sizes of chaos, we shall also present, in the context of Koopman operators on Lp spaces, 1 ≤ p < ∞, with dissipativity and bounded distortion, new characterizations of weakly mixing and frequent hypercyclicity. This will allow us, as a consequence, to deduce analogous conclusions for fundamental mathematical objects: bilateral weighted backward shifts on l_p spaces

Date. March 31, 14h00 

Speaker   Carllos Eduardo HOLANDA (ICMC-USP and LED-UFAL)  

Title.  Some relations between additive and nonadditive sequences of potentials

Abstract.  

Based on recent developments, we will talk about some relations between additive, almost additive and asymptotically additive sequences of continuous functions. In addition, we will see how these relations can affect the nonadditive thermodynamic formalism and multifractal analysis for maps and flows. 

Date. March 03, 14h00 (UTC/GMT+1)

Speaker   Yun ZHAO (Suzhou University)

Title.   Measures of maximal and full dimension for smooth maps

Abstract.  

For a C^1 non-conformal repeller, we will show that there exists an ergodic measure of full Caratheodory singular dimension. For an  average conformal hyperbolic set of a C^1 diffeomorphism, we construct a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a C^{1+\alpha} diffeomorphism, we will show that there exists an ergodic measure of maximal dimension.

Zoom link:
https://videoconf-colibri.zoom.us/j/91542474700?pwd=S0hGRHdtTGlucHo1SStJR1Vib1FTZz09

Meeting ID: 915 4247 4700

Password: 254338

Date. February 17, 14h00 (UTC/GMT+1)

Speaker   Stephen CANTRELL (University of Chicago)

Title.  Orbital counting for Green metrics

Abstract.  Suppose we run a random walk on a group. The corresponding Green metric assigns distances between group elements as follows: the distance between group elements x and y is the negative logarithm of the probability that, when we start our random walk at x, we reach y. In this talk we'll discuss how to obtain orbital counting results for Green metrics by using techniques from ergodic theory.

Zoom link:
https://videoconf-colibri.zoom.us/j/94104329635?pwd=dlFFNlRhazhXRFpEQ0UrMCtlL1U1QT09

Meeting ID: 941 0432 9635

Password: 859096

Date. February 03, 14h00 (UTC/GMT+1)

Speaker   David BURGUET (Université Paris 6)

Title.  Topological mean dimension of induced systems

Abstract.  

The topological mean dimension is a topological invariant introduced by Gromov, which is zero for topological systems with finite dimension or finite topological entropy.  For a topological system (X,T), we consider the induced map $T_*$ on the set $\mathcal M(X)$ of Borel probability measures. It is well known that $T_*$ has infinite topological entropy, if  $T$ has positive topological entropy. Also, whenever $X$ is infnite, the simplex $\mathcal M(X)$ has infinite topological dimension. We show that the topological mean dimension of $T_*$ is also infinite. Morevoer we give precise rates of divergence  of $h_W(T_*,\epsilon)$ when $\epsilon$ goes to zero, where $h_W(T_*,\epsilon)$ denotes the Bowen metric entropy with respect to the Wasserstein distance $W$. The proof uses the independence theory developped by Glasner-Weiss, Kerr-Li and others.
  Joint work with Ruxi Shi.

Date. December 09, 14h00 (UTC/GMT+1)

Speaker   Ale Jan HOMBURG (KdV Institute for Mathematics - University of Amsterdam)

Title.  

Iterated function systems of linear expanding and contracting maps on the unit interval  

Abstract.  

We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers.  This dynamics depends on the Lyapunov exponent.  For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existence of an absolutely continuous stationary measure for the two-point dynamics and discuss its consequences.  For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicatively dependent integers, we provide explicit expressions for absolutely continuous stationary measures of the two-point dynamics. These stationary measures are infinite $\sigma$-finite measures in the case of zero Lyapunov exponent.  This is joint work with Charlene Kalle.

Date. November 25, 14h00 (UTC/GMT+1)

Speaker   Magdalena FORYS-KRAWIEC

Title.  Irregular sets for maps with shadowing and their metric mean dimension

Abstract.  

For a dynamical system, a Φ-irregular set is the set of points x for which the Birkhoff averages determined by Φ along the orbit of x diverge. Such sets, however negligible from ergodic theory point of view, may have interesting dynamical properties and complex structure. In the talk we present the characterization of Φ-irregular sets in dynamical systems with shadowing over a chain recurrent class in terms of their metric mean dimension. The results presented in the talk are obtained as a joint work with Piotr Oprocha.

Date. November 11, 14h00 (UTC/GMT+1)

Speaker   Zemer KOSLOFF (Hebrew University of Jerusalem)

Title.  (Stationary) Bernoulli shift factors for inhomogeneous systems

Abstract.  

Joint work with Terry Soo. We will survey recent results on the model of nonsingular Bernoulli shifts, its relation to classical questions in smooth ergodic theory and discuss recent results on existence of (stationary) Bernoulli shift factors for systems which are not measure preserving and the stark difference between finitary and non-finitary factors.

Date. October 28, 14h00 (UTC/GMT+1)

Speaker   Aihua FAN  (Université de Picardie Jules Verne, Amiens)

Title.  Some aspects of weighted Birkhoff averages: Bohr Chaoticity and Multifractality

Abstract.  

The Sarnak conjecture, which concerns with the Birkhoff averages  weighted by the Möbius sequence in a topological dynamical system,  asserts that all zero entropy systems are  orthogonal to the Möbius  sequence. Which systems are orthogonal to none of non-trivial  weighted? We qualify such systems as Bohr chaotic. The Bohr chaoticity  is a complexity measure and is a topological invariant; it implies the  positivity of entropy. However, the positivity of entropy doesn’t  imply the Bohr chaoticity. We prove that a system (X, T) admitting a  horseshoe (i.e a susbsytem of some power of T is conjugate to a full  shift) is Bohr chaotic. Thus usual nice systems of positive entropy are Bohr  chaotic. But systems having few ergodic measures are not Bohr chaotic.    Another class of systems which are proved to be Bohr chaotic are the algebraic principal systems.  A second aspect of weighted  Birkhoff averages to be discussed will be its multifractal behaviors.  To this end, a thermodynamical formalism is proposed, but Gibbs  measure in this new setting are not ergodic with respect to the  original dynamical system.  In the computation of the multifractal  spectrum, it arises a question about a topological version of  Furstenberg-Kesten theorem on products of non-negative matrices and a  such version is obtained.  Parts of results presented here are joint works with Shilei FAN  (Wuhan), Valery RYZHYKOV (Moscou), Klaus SCHMIDT (Vienne), Weixiao  SHEN (Shanghai) et Evgeny VERBITSIY (Leiden), Meng WU (Oulu); others  are due to Matan TAL (Jerusalem), Balazs BARANY (Budapest), Michal  RAMS (Warsaw), Ruxi SHI (Paris).

Zoom link: https://videoconf-colibri.zoom.us/j/85292642527?pwd=GqAJL2PfDJbuVAqQ1WT2QBgO-w-iEG.1

Meeting ID: 852 9264 2527

Password: 576962

Date. October 14, 14h00 (UTC/GMT+1)

Speaker   Joel MOREIRA (Warwick)

Title.  Infinite Ergodic Ramsey theory

Abstract.  

In the 1970's Furstenberg obtained an ergodic theoretic proof of Szemeredi's theorem, stating that any set of natural numbers with positive upper density contains arbitrarily long (but finite) arithmetic progressions. Since then, Furstenbeg's approach has been extended to many other combinatorial applications, giving rise to the field of Ergodic Ramsey Theory. However, until recently, ergodic theoretic methods were unable to handle infinite configurations. I will present recent joint work where we overcame this difficulty to answer some Ramsey theoretical questions raised by Erdos in the 1980's using methods from ergodic theory.

Zoom Link: https://videoconf-colibri.zoom.us/j/88629332593?pwd=hp69glSv748aFo74aJs8c5YromEwKE.1

Meeting ID: 886 2933 2593

Password: 373077

Date. September 30, 14h00 (UTC/GMT+1)

Speaker   Charlene KALLE (University of Leiden)

Title.  Random intermittent dynamics

Abstract.   Intermittent dynamics, where systems irregularly alternate between long periods of different types of dynamical behaviour, has been studied since the work of Pomeau and Manneville in 1980. In random dynamical systems this phenomenon has only been well understood in a few specific cases. A random dynamical system consists of a family of deterministic systems, one of which is chosen to be applied at each time step according to some probabilistic rule. In this talk we will describe the intermittency of some families of random systems with a particular emphasis on how the intermittency of the random system depends on the intermittency of the underlying deterministic systems. This talk is based on joint works with Ale Jan Homburg, Tom Kempton, Valentin Matache, Marks Ruziboev, Masato Tsujii, Evgeny Verbitskiy and Benthen Zeegers.

Date. September 16, 14h00 (UTC/GMT+1)

Speaker   Sakshi JAIN (Università degli Studi di Roma - Tor Vergata)

Title.  Discontinuities cause essential spectrum

Abstract.  

 We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large for any relevant Banach space when the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to the theoretical best possible case.

Date. June 17, 14h00 (UTC/GMT+1)

Speaker   Jorge FREITAS (Universidade do Porto)

Title.  Point processes, record events and heavy tails 

Abstract.  We consider stochastic processes arising from chaotic systems and build multidimensional point processes  which provide a nice description regarding both the extremal behaviour, the distribution of record events and the convergence of functional limits for the sums of heavy tailed observables.

Date. June 17, 14h50 (UTC/GMT+1)

Speaker  Minsung KIM

Title.  Deviation spectrum of  Birkhoff integrals for locally Hamiltonian flows on compact surfaces.  

Abstract.  

This talk will consist of an introduction to the topic of deviation of Birkhoff integrals for locally Hamiltonian flows on compact surfaces and recent related results. These works extend the study of the spectrum of deviations of ergodic integrals beyond the case where the observable vanishes at the singularities. New developments include a better understanding of the asymptotics (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime). This is joint work with Krzysztof Frączek.

15 minutes break

Date. June 17, 15h50 (UTC/GMT+1)

Speaker  Tanja SCHINDLER

Title. A qualitative central limit theorem for certain unbounded over piecewise expanding interval map

Abstract.  

Many limit theorems in ergodic theory are proven using the spectral gap method. So one of the main ingredients for this method is to have a space on which the transfer operator has a spectral gap. However, most of the classical spaces, like for example the space of Hölder or quasi-Hölder function or BV functions don't allow unbounded functions. We will give such a space which allows observables with a pole at the fixed points of a piecewise expanding interval transformation and state a quantitative central limit theorem using Edgeworth expansions. As an application we give a sampling result for the Riemann-zeta function over a Boolean type transformation. This is joint work with Kasun Fernando.

Date. June 03, 14h00 (UTC/GMT+1)

Speaker   Adriana SANCHEZ (CIMPA - Universidad de Costa Rica) 

Title.  Analiticity of the Lyapunov exponents of random products of quasi-periodic cocycles

Abstract.  

One of the most important questions that comes up when studying Lyapunov exponents of linear cocycles is the regularity problem. By this we mean the problem of understanding the behavior of the Lyapunov exponents regarding the underlying data. It is a classical result of Peres [1] that, for i.i.d. sequence of matrices with discrete distribution, the Lyapunov exponents are locally analytic if mild simplicity conditions are assumed. In this talk we are going to discuss an extension of Per ́es conclusions in [1] to include the random product of quasi-periodic cocycles for the skew product base dynamics. This is a join work with Jamerson Bezerra and El Hadji Yaya Tall.

[1] Y. Peres. “Analytic dependence of Lyapunov exponents on transition probabilities”. In: Lyapunov exponents (Oberwolfach, 1990). Vol. 1486. Lecture Notes  in Math. Springer-Verlag, 1991, pp. 64–80.

Date. June 03, 15h00 (UTC/GMT+1)

Speaker  Emma D'ANIELLO  (Università degli Studi della Campania "Luigi Vanvitelli")

Title.  Some linear dynamics on Lp spaces 

Abstract.  

We investigate, in the linear setting, notions of chaos coming from topological dynamics, and hyperbolic properties originally defined in a non-linear framework.  We focus in particular on “shift-like” composition operators on arbitrary measure spaces, a large class of linear operators that includes the weighted shifts as a special case. 

Zoom link: 

https://videoconf-colibri.zoom.us/j/87877112841?pwd=aHBRd21HSUZkM3Z5QXRpeUhIZWl2dz09

Meeting ID: 878 7711 2841
Password: 654122

Date. May 20, 14h00 (UTC/GMT+1)

Speaker   Simone CERREIA-VIOGLIO (Bocconi)

Title. Ergodic Theorems for Lower Probabilities

Abstract.  We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers (joint work with F. Maccheroni and M. Marinacci).

Date. May 20, 15h00 (UTC/GMT+1)

Speaker  Héctor BARGE (Universidad Politécnica de Madrid)

Title.  Low-dimensional topology and the realization problem of attractors for non-connected compacta

Abstract.  

It is well known that if K = K_1 ⊔ . . . ⊔ K_r ⊂ R^n, where K_i is compact for each i, is an attractor for a dynamical system (discrete or continuous) in R^n then K_i is an attractor for each i. In this talk we shall study whether the converse statement holds:

(S): If each K_i is an attractor for a dynamical system so is K.

We shall see that (S) holds true in the case of continuous dynamical systems. However, the discrete case is more involved. In particular, we will see that there exist compacta in R^3 comprised of two connected components that can be realized as attractors separately but whose union cannot be so realized. Finally, we will introduce a sufficient condition for (S) to hold true in the discrete case and we shall see that this condition is also sufficient in some suitable situations. These results have been obtained in collaboration with J.J. Sanchez-Gabites.

Date. May 06, 14h00 (UTC/GMT+1)

Speaker   Disheng XU (Beijing University)

Title. Some topics on compact center foliations of partially hyperbolic systems

Abstract.  

Partially hyperbolic systems is a natural generalization of Anosov systems in dynamical systems. In this talk we will focus on partially hyperbolic systems with invariant compact center foliations, and discuss some interesting properties and some recent results on the regularity of compact center foliations, under different conditions. This talk will be self-contained.

Date. May 06, 15h00 (UTC/GMT+1)

Speaker   Luciana SALGADO (Universidade Federal do Rio de Janeiro)

Title.  Ergodic and geometric aspects of  singular flows

Abstract.  

Lyapunov functions provided by quadratic forms are useful tools  in Dynamical Systems. For instance, these are used in results of topological stability and ergodic properties as found in works by Burns-Katok,  Lewowicz, Markarian and Wojtkowski. This talk is about how to use the quadratic Lyapunov functions to show some ergodic and dynamical properties of singular flows (e.g. Lorenz flow).

Date. April 22, 14h00 (UTC/GMT+1)

Speaker   Adam SPIEWAK (Bar-Ilan University)

Title. Typical absolute continuity for classes of dynamically defined measures

Abstract.  

Consider a one-parameter family of iterated function systems on the interval and a family of measures on the corresponding symbolic space, depending on the same parameter. We study geometric properties of measures projected by the natural projection corresponding to the IFS.  Assuming that IFS satisfies a transversality condition and measures depend on the parameter regularly enough, we obtain results on the value of the dimension and absolute continuity of the projected measure for almost every parameter. This extends classical results (for projections of a fixed measure) and applies e.g. to invariant measures for systems with place-dependent probabilities and equilibrium measures for hyperbolic iterated function systems. The talk is based on a recent joint work with Balázs Bárány, Károly Simon and Boris Solomyak.

Date. April 22, 15h00 (UTC/GMT+1)

Speaker  Agnieszka ZELEROWICZ (University of Maryland) 

Title. Lorentz gases on aperiodic tilings

Abstract.  

 In this joint work with Rodrigo  Treviño we consider the Lorentz gas model of category A (that is, with no corners and of finite horizon) on aperiodic repetitive tilings of $\mathbb{R}^2$ of finite local complexity. We show that the compact factor of the collision map has the K property, from which we derive mixing for pattern-equivariant functions as well as the planar ergodicity of the Lorentz gas flow.

Zoom link: https://videoconf-colibri.zoom.us/j/88664008491?pwd=MVdudWFaVFJkVzk1SlNKdjQ0b01Fdz09

Meeting ID: 886 6400 8491
Password: 732004

Date. April 08, 14h00 (UTC/GMT+1)

Speaker   Tomoo YOKOYAMA (Gifu University)

Title.  Existence and Non-existence of Length Averages for Foliations

Abstract.  

There are concepts and methods of dynamical systems theory that are applied to foliations. One of the pioneering works was done by Ghys et al. [1], in which entropies of foliations were first introduced (a close link between the entropy of foliations). This talk introduces "time average" for foliation, gives examples with/without the averages, and demonstrates that the averages exist everywhere for any codimension one singular foliations on compact surfaces under a mild condition in strong contrast to time averages surface flows. In other words, we aim to investigate the complexity of foliations by generalizing the existence problem of time averages in dynamical systems theory to foliations: It has recently been realized that a positive Lebesgue measure set of points without time averages only appears for complicated dynamical systems, such as dynamical systems with heteroclinic connections or homoclinic tangencies [2]. This work is based on [3] and is joint work with Yushi Nakano. 

[1] Ghys,É., Langevin,R., Walczak,P. Entropie géométrique des feuilletages. Acta Math.160(1),105–142 (1988)
[2] Kiriki, S., Soma, T.: Takens’ last problem and existence of non-trivial wandering domains. Adv. Math. 306, 524–588 (2017)
[3] Nakano, Y., Yokoyama, T. Existence and non-existence of length averages for foliations. Comm. Math. Phys. 372 (2019), no. 2, 367–383.

Date. April 08, 15h00 (UTC/GMT+1)

Speaker  Sebastién ALVAREZ (Universidad de la República)

Title. Metric entropy of foliations

Abstract.  

The topological dynamiucs of foliations is wll understood. There is an analogue of periodic orbits, limit sets, recurrence, and even topological entropy, following the work of Ghys, Langevin and Walczak. This entropy measures the transverse separation of the leaves of a foliation. This is not the case of the ergodic theory of foliations. In particular, there is no satisfactory version of metric entropy for foliations, and it is difficult to imagine a way to detect the separation of leaves using measure theory (whether harmonic, or invariant by certain dynamics tangent to the leaves). For example is it possible to obtain a variational principle for foliations? In this talk, I will discuss an approach made with Jiagang Yang (UFF, Niteroi) to attack this problem.

Zoom link:  https://videoconf-colibri.zoom.us/j/84364028076?pwd=WFBFSlRiMEdOOFBLbE84ZXJTOW9LQT09 

Meeting ID: 843 6402 8076
Password: 617138

Date. March 18, 14h00 (UTC/GMT+0)

Speaker   Simon BAKER (University of Birmingham)

Title.  Normal numbers and self-similar measures

Abstract.  

A real number x is said to be normal in base b if the sequence (b^n x) is uniformly distributed modulo one. In this talk I will discuss a recent result which states that for a self-similar measure and an integer b, if the self-similar measure and b satisfy a suitable arithmetic assumption, then almost every x is normal in base b with respect to the self-similar measure. This is a joint work with Amir Algom and Pablo Shmerkin.

Date. March 18, 15h00 (UTC/GMT+0)

Speaker  Danijela DAMJANOVIC (KTH) 

Title. Exploring the Borel-Smale example: centralizer rigidity for maps and center rigidity for the action

Abstract.  

 I will talk about our exploration of an interesting Z^2 Anosov action on a 2-step nilmanifold.  Historically, Anosov elements of this action were presented by Smale as first examples of Anosov maps which are not on the torus. We provide classification of centralizers for perturbations of certain non-Anosov elements of this action. We also show that the action itself is not locally rigid, but it still enjoys certain center-rigidity property under perturbations. This is joint work with Amie Wilkinson and Disheng Xu.

Date. March 04, 14h00 (UTC/GMT+0)

Speaker  Fagner B. RODRIGUES (Universidade Federal do Rio Grande do Sul)

Title.  Properties of the set of entropy points of a free semigroup action

Abstract.  

The aim of this talk is to present some local properties of the topological entropy of a free semigroup action. In order to do that we focus on the set of entropy points of a free semigroup action, show that this set carries the full entropy of the system (which, with respect to the chaocity of the system, gives a fundamental relevance to such set). We prove that the topological entropy of a free semigroup action focuses on a countable set of points.   This is a joint work with Marcus V. Silva and Thomas Jacobus, both are PhD students of the PhD program of Mathematics and Statistics  Institute of the Federal University of Rio Grande do Sul. 

Date. March 04, 15h00 (UTC/GMT+0)

Speaker  Jamerson BEZERRA (Copernicus University in Toruń)

Title. Regularity of the Lyapunov exponent of random product of matrices

Abstract.  

It is well established in the literature that the Lyapunov exponent of the random product of matrices varies continuously with respect to the underlying data. However, the understanding of the right modulus of continuity of this function remains an active problem with many contributions in the past years.  In this talk we present an upper bound for the regularity of the Lyapunov exponent of random product of (generic) matrices. This upper bound can be characterized in terms of metric properties of another essential invariant of the system namely the stationary measure. 

Date. February 18, 14h00 (UTC/GMT+0)

Speaker  Martin ANDERSSON (Universidade Federal Fluminense)

Title. Non-uniformly expanding maps in the "wrong" homotopy classes

Abstract.  

Examples of non-uniformly expanding maps are usually constructed by deforming a uniformly expanding map along a (locally supported) homotopy. In this talk I intend to explain how to create new examples of non-uniformly expanding maps by deforming other kinds of maps, including Anosov endomorphisms.

Date. February 18, 15h00 (UTC/GMT+0)

Speaker  Katsutoshi SHINOHARA (Hitotsubashi University & Université de Bourgogne) 

Title. On A/D-converters and Fredholm determinants of piecewise linear maps. 

Abstract.  

Beta-encoders are A/D (Analog-to-Digital) encoders based on beta-expansions. In order to give estimates of their quantization errors theoretically, the spectral analysis of Perron-Frobenius operators of corresponding piecewise linear maps, in other words, the study of dynamical zeta functions, plays an important role. I will give an introductory explanation of this interplay between electronic circuit design theory and the theory of one-dimensional dynamical systems.  

Reference: Estimation of mean squared errors of non-binary A/D-encoders through Fredholm determinants of piecewise-linear transformations https://doi.org/10.1587/nolta.9.243

Zoom link: https://videoconf-colibri.zoom.us/j/82241496078?pwd=T2lnZG1FUmpXUjFVdTdwZ3R6V1Vndz09

Meeting ID: 822 4149 6078
Password: 198373

Date. February 04, 14h00 (UTC/GMT+0)

Speaker  Paolo GIULIETTI (Università di Pisa)

Title. Quenched linear response for smooth expanding on average cocycles

Abstract.  

I will present  an abstract quenched linear response result for random dynamical systems, which applies to the case of smooth expanding on average cocycles on the unit circle. Confronting with the existing results in the literature,  the class of random dynamics studied does not necessarily exhibit uniform decay of correlations.  Joint work with D. Dragicevic (Univ. of Rijeka)  and J. Sedro (Sorbonne Univ.)

Date. February 04, 15h00 (UTC/GMT+0)

Speaker  Anna GIORDANO BRUNO ( Università degli Studi di Udine)

Title. Topological and algebraic entropy on LCA groups

Abstract.  

 Since its origin, the algebraic entropy h_alg was introduced in connection with the topological entropy h_top by means of the Pontryagin - van Kampen duality. For a continuous endomorphism φ: G → G of a locally compact abelian group G, denote by \hat G the Pontryagin dual group of G and by \hat φ: G → G the dual endomorphism of φ. In this talk, we see under which assumptions the equality h_top(φ) = h_alg(\hat φ) is known to hold. We present the same connection also in the much more general setting of (semi)group actions.

Link: https://videoconf-colibri.zoom.us/j/87241036903?pwd=b3Y2S1lLS1QwanFTbTNSdmR6Z25IUT09

Meeting ID:  872 4103 6903
Password:  628922

Date. January 21, 14h00 (UTC/GMT+0)

Speaker  Cristina LIZANA (Universidade Federal da Bahia)

Title.  Invariance of entropy for maps isotopic to Anosov

Abstract.  

We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of T^d with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on H_1(T^d) is hyperbolic. In absence of the simplicity condition we construct a robustly transitive counter-example. This is a joint work with P. Carrasco, E. Pujals and C. Vásquez.

Date. January 21, 15h00 (UTC/GMT+0)

Speaker  Çağrı SERT (Universität Zürich)

Title. Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Abstract.  

 Let $\Gamma$ be a hyperbolic group and S a finite symmetric generating set. The choice of S determines a metric on $\Gamma$ (namely the graph metric on the associated Cayley graph). Given a representation $\rho: \Gamma \to \GL_d(\R)$, we are interested in obtaining results analogous to random matrix products theory (RMPT) but for the deterministic sequence of spherical averages (with respect to S-metric). We will discuss a general law of large numbers and more refined limit theorems such as central limit theorem and large deviations. If time allows, we will also see boundary limit theorems and convergence of interpolated matrix norms along geodesic rays to the standard Brownian motion. The techniques involve the existence of a coding for hyperbolic groups, RMPT in Markovian dependence and ergodicity of Patterson--Sullivan measures. These as well as the connections with (and results in) the classical RMPT will be discussed. Joint work with S. Cantrell.

Link:  https://videoconf-colibri.zoom.us/j/84335110437?pwd=VVlkcExHSFVTWmxkZTlHQ2ozY2lFZz09
Meeting ID:  843 3511 0437
Password: 055613 

Date. December 17, 14h00 (UTC/GMT+0)

Speaker  Antti KÄENMÄKI  (University of Oulu)

Title. Finer geometry of planar self-affine sets

Abstract.  For a planar self-affine set satisfying the strong separation condition, it has been recently proved that under mild assumptions the Hausdorff dimension equals the affinity dimension. In this article, we continue this line of research, and our objective is to acquire more refined geometric information in this setting. In a large class of such non-carpet planar self-affine sets, we characterize Ahlfors regularity, determine the Assouad dimension of the set and its projections, and estimate the Hausdorff dimension of slices. We also demonstrate that the Assouad dimension is not necessarily bounded above by the affinity dimension. The talk is based on a recent work with Balázs Bárány and Han Yu.

Date. December 17, 15h00 (UTC/GMT+0)

Speaker   Benoit SAUSSOL (Université de Bretagne Occidentale)

Title.  Mikado of geodesics on negatively curved manifolds

Abstract.  Recently Athreya, Lalley, Sapir and Wroten have been interested in the tangle of geodesics in a compact riemaniann surface of negative curvature. One question is to understand locally the picture of a geodesic segment of length T, in a vicinity of  any point given on the surface. With Françoise Pène we recover their main result, applying our work on spatio-temporal point processes for visits to small sets. 

Zoom link (session will open some minutes before 14h00):
https://videoconf-colibri.zoom.us/j/87853308336?pwd=UTdiRDBTR08yMjF1YjhQWHdldnpQQT09

Meeting ID: 878 5330 8336
Password: 660164

Date. December 03, 14h00 (UTC/GMT+0) 

Speaker  Julien SEDRO (LPSM-Paris) 

Title. Fractional susceptibility functions for the quadratic family

Abstract. In the last two decades, a clear picture of linear response for one-dimensional dynamics emerged. However, in the specific instance of the quadratic family, two seemingly contradictory results coexist: Ruelle proved in 2005 that the formal candidate for the derivative of the response function at 0 was well defined, through the study of the susceptibility function, while Baladi et al. proved in 2014 that the same response function was at best 1/2 Hölder at 0.  To try to "shed some light on this puzzling state of affairs", Baladi and Smania introduced so called fractional susceptibility functions, which are complex functions of two complex variables, and relate to fractional response in the same way the susceptibility function relates to linear response. I will present results establishing holomorphy on a disk of radius larger than one for simplified fractional susceptibility functions, as well as explain the strategy to extend this result to the 'true' fractional susceptibility function.

Date. December 03, 15h00 (UTC/GMT+0)

Speaker  Barbara SCHAPIRA (Université de Rennes)

Title.  Strong positive recurrence and applications

Abstract.  In a series of works with S Tapie, R Coulon, R Dougall, S Gouezel about the ergodic properties of the geodesic flow on noncompact negatively curved manifolds, we introduced and used notions of entropy and pressure at infinity, and of strong positive recurrence (when entropy / pressure at infinity is strictly smaller than the topological entropy / pressure). I will present these notions and some applications.

Date. November 19, 14h00 (UTC/GMT+0)

Speaker  Giovanni PANTI (Università Degli Studi Di Udine)

Title.  Attractors of dual continued fractions

Abstract.

We identify continued fractions with piecewise-projective Markov maps on some  interval, provided that all branches are induced by elements in a fixed Hecke triangle group. Ordinary continued fractions constitute a very special case, relative to the (2,3,infinity) group. The branches of the map are expanding on the domain interval, but contracting on some other part of the real projective line. As such, they give rise to an IFS whose attractor is relevant, since it determines the natural extension and several properties, both algebraic and dynamical, of the original system. We use an appropriate generalization of the classical Minkowski Question Mark function to simultaneously linearize all maps resulting from the same triangle group, and to prove that the above IFS satisfies the open set condition. We draw consequences -as well as open problems- from these facts.

Date. November 19, 15h00 (UTC/GMT+0)

Speaker   Daniel SMANIA (ICMC-USP-São Carlos)

Title.  Birkhoff sums as distributions

Abstract.

Under conditions (sufficiently fast decay of correlations)  that are satisfied in various settings we can consider Birkhoff sums as distributions in the Schwartz sense (as in the study of PDEs) and study their regularity. We will do this study in the case of  piecewise expanding one-dimensional maps and some other cases. This study makes intensive use of the statistical properties of the dynamic systems under consideration (decay of correlations, central limit theorem). In addition to its intrinsic interest, this study is motivated by the theory of deformations of dynamical systems, something similar to Teichmuller's theory for compact Riemann surfaces, but in infinite dimension, which has applications in important questions about the perturbation of dynamical systems, such as the problem of linear response. Indeed, in  one-dimensional dynamics topological classes often have a differentiable structure, that is, they are infinite-dimensional Banach manifolds, and the study of Birkhoff sums as distributions has a surprisingly important role here. Joint work with Clodoaldo Ragazzo (IME-USP), but we will also cite several results with Viviane Baladi and Amanda de Lima.

Zoom link (session will open some minutes before 14h00):
https://videoconf-colibri.zoom.us/j/81742604247?pwd=SGtSTTdwL2Z4OE40dndPSVVha3VPZz09

Meeting ID: : 817 4260 4247
Password:  926354

Date. November 05, 14h00 (UTC/GMT+0)

Speaker  Olga LUKINA (University of Vienna)

Title. Weyl groups and the growth of cycles in actions on rooted trees

Abstract.

Maximal tori and Weyl groups play an important role in the study of compact connected Lie groups. In this talk, we introduce an analogous notion for profinite groups acting on rooted trees. We then use Weyl groups to study the conjecture by Boston and Jones, that settled elements are dense in the images of representations of absolute Galois groups of number fields into automorphism groups of rooted trees. An element is settled if it satisfies certain conditions on the growth of cycles in its restrictions to finite levels of the tree. This is joint work with Maria Isabel Cortez.

Date. November 05, 15h00 (UTC/GMT+0)

Speaker   Pablo CARRASCO (Universidade Federal de Minas Gerais)

Title. Non-uniformly hyperbolic endomorphisms

Abstract.

Consider an endomorphism E of the 2-dimensional torus (a matrix with integer coefficients and determinant equal to one) and suppose that it is not a multiple of the identity. Then there exists an endomorphism homotopic to E that is area preserving and which is robustly (C1!) non-uniformly hyperbolic; it has one positive and one negative Lyapunov exponent, almost everywhere with respect to Lebesgue.  This result is part of a joint work with Martin Andersson (UFF) and Radu Saghin (PUCV).

Date. October 22, 14h00 (UTC/GMT+1)

Speaker  Ali TAHZIBI (ICMC-USP)

Title.  Equilibrium states for Discretized Anosov flows  

Abstract.

We consider partially hyperbolic diffeomorphisms which appeared in the classification of partially hyperbolic diffeomorphisms homotopic to identity on three-manifolds. In a previous result with Buzzi and Fisher we constructed measures of maximal entropy for such diffeomorphisms and later proved some rigidity result about such measures joint with Buzzi-Crovisier-Poletti. In this talk we announce also some existence/construction results and discuss uniqueness of (u-)equilibrium states  in this setting in a work that is essentially part of the thesis of my student Richard Becerra.

Date. October 22, 15h00 (UTC/GMT+1)

Speaker   Tamara KUCHERENKO (CCNY)

Title. Flexibility of the Pressure Function

Abstract.

 We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics, since they correspond to phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We prove that in a setting of one-dimensional compact symbolic systems these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter.  As a consequence, we obtain that for a continuous observable the phase transitions  can occur at a countable dense set of temperature values. We go further and show that one can vary  the cardinality of the set of ergodic equilibrium states as a function of the parameter to be any number,  finite or infinite.​ This is based on joint work with Anthony Quas.

Date. October 08, 14h00 (UTC/GMT+1)

Speaker  Catalina FREIJO (Lisbon)

Title. Lyapunov exponents for non-uniformly fiber bunched linear cocycles.

Abstract.

We consider a fixed hyperbolic dynamics in the base and perturb the cocycle for studying the continuity of Lyapunov exponents. For the case of uniformly fiber bunched cocycles, continuity was proved by Backes, Brown and Butler. In this work we present a characterization for continuity in the case of non-uniformly fiber bunched dynamics which is a partial answer to the conjecture of Marcelo Viana. This is a joint work with Karina Marin (UFMG).

Date. October 08, 15h00 (UTC/GMT+1)

Speaker  Silvius KLEIN (PUC-Rio de Janeiro)

Title.  Mixed random-quasiperiodic cocycles  

Abstract. 

The purpose of this talk is to introduce the concept of mixed random-quasiperiodic linear cocycles and to discuss the ergodicity of its base dynamics as well as the positivity and  continuity of the corresponding Lyapunov exponent. This is part of a larger joint project with Ao Cai and Pedro Duarte (both from the University of Lisbon) whose aim is to study the stability under random perturbations of certain quasiperiodic systems.

Date. September 24, 14h00 (UTC/GMT+1)

Speaker  Kenichiro YAMAMOTO (Nagaoka)

Title. Large deviation principle for piecewise monotonic maps with density of periodic measures

Abstract. 

We show that a piecewise monotonic map with positive topological entropy satisfies the level-2 large deviation principle with a unique measure of maximal entropy as reference under the conditions that the corresponding Markov diagram is irreducible and that the periodic measures of the map are dense in the set of ergodic measures. This result can apply to a broad class of piecewise monotonicmaps, such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces.

Date. September 24, 15h00 (UTC/GMT+1)

Speaker  Dario DARJI (University of Louisville) 

Title.  Generalized Hyperbolicity and the Shadowing Property in Linear Dynamics

Abstract.

In this talk we discuss recent results  and open questions  concerning in Hyperbolicity, Generalized Hyperbolicity and the Shadowing Property in the setting of linear dynamics. In particular, we discuss some results concerning when the Generalized Hyperbolicity is equivalent to the Shadowing Property. This talk will be accessible to grad students and non-specialists. All notions and background material will be clearly explained. 

Date. September 10, 14h00 (UTC/GMT+1)

Speaker  Sylvain CROVISIER (Université Paris-Saclay)

Title.  Strong positive recurrence for surface diffeomorphisms

Abstract.

We introduce the strong positive recurrence property for diffeomorphisms of a manifold and show that it holds for every C-infinity surface diffeomorphism with positive topological entropy. It implies that the ergodic measures maximizing the entropy have exponential decay of correlations and satisfy a central limit theorem. This is a joint work with J. Buzzi and O. Sarig.

Date. September 10, 15h00 (UTC/GMT+1)

Speaker  Godofredo IOMMI (PUC-Chile) 

Title. An ergodic theorem for symmetric averages

Abstract.

We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As a consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converges to the Halász-Székely barycenter of the corresponding distribution. This is joint work with Jairo Bochi and Mario Ponce.

Date. July 09 

14h00-14h50 (UTC/GMT+1)

Speaker.  Rafael POTRIE (Universidad de la Republica)

Title. Ergodicity of a class of partially hyperbolic systems

Abstract.

We study the problem of determining when a partially hyperbolic diffeomorphism in dimension 3 is ergodic without need for perturbations. We show that for a large class of such systems, that we called collapsed Anosov flows and which cover full connected components of partially hyperbolic systems we have ergodicity unless there is a torus tangent to the $su$-direction which is known to impose strong conditions on the topology of the underlying manifold. I will try to explain the context of this result, and hopefully be able to provide some taste of the proofs. This is joint work with S. Fenley. 

15h00-15h50 (UTC/GMT+1)

Speaker.  Jerôme ROUSSEAU (Federal University of Bahia / University of Porto)

Title. Longest common substring for random subshifts of finite type

Abstract.

We study the behaviour of the longest common substring for random subshifts of finite type. We prove that, under some exponential mixing assumptions, this behaviour is linked to the Rényi entropy. We will show quenched and annealed results. In particular, we will see that in the quenched case we obtain a phase transition which does not appear in the annealed case. This talk includes some joint works with Adriana Coutinho, Rodrigo Lambert, Sébastien Gouezel and Manuel Stadlbauer.

16h10-17h00 (UTC/GMT+1)

Speaker.  Juan RIVERA-LETELIER (University of Rochester)

Title. Prime orbit counting for Collet-Eckmann maps

Abstract.

A discussion of prime orbit counting results for real and complex maps, where periodic orbits are weighted according to their Lyapunov exponents. For real and complex maps satisfying the Collet-Eckmann condition, we obtain a result with a power savings error term. In comparison with the hyperbolic case, the main technical difficulty stems from the relatively wild geometric properties of the absolutely continuous invariant measure. This is a joint work with Li Zhiqiang. 

Date. July 02, 14h00 (UTC/GMT+1)

Speaker  Davor DRAGICEVIC (University of Rijeka)

Title. Quenched limit theorems for expanding on average cocycles

Abstract.

In [1, 2], the authors extended the classical spectral method for establishing limit theorems for deterministic dynamical systems to random piecewise-expanding and hyperbolic dynamical systems exhibiting uniform decay of correlations. In this talk we will discuss a certain modification of this method that enables us to establish limit theorems for a class of random piecewise expanding dynamics that doesn't exhibit uniform decay of correlations. This is a joint work with Julien Sedro [3].  

[1] D. Dragičević et. al. A spectral approach for quenched limit theorems for random expanding dynamical systems, Comm. Math. Phys. 360 (2018), 1121--1187.
[2] D. Dragičević et. al. A spectral approach for quenched limit theorems for random hyperbolic dynamical systems, Trans. Amer. Math. Soc. 373 (2020), 629--664.
[3] D. Dragičević and J. Sedro, Quenched limit theorems for expanding on average cocycles, preprint (2021), https://arxiv.org/abs/2105.00548

A conference on Dynamical Systems
Homepage: https://ergwork.web.unc.edu/

2021 Chapel Hill Ergodic Theory Workshop : Global Online Edition

 The activities of the Virtual Ergodic Conference will take place in three consecutive weeks, from June 21 to July 09. While schedules are thought to adapt most time zones, these are specially fitted to North & South America (week 1), Europe & Africa (week 2) and Asia & Australia (week 3). The format will try to preserve the atmosphere of the Chapel Hill Ergodic Theory Workshops with some adaptations to match current virtual modalities.

The second week of the workshop will take place on Monday, June 28th and Tuesday, June 29th.
Schedule (all times are CEST):

Monday June 28
13:30-14:00  - Zoom room opens
14:00-14:50  - Mark Pollicott
15:00-15:20  - Michael Lin
15:30-15:50  - Guy Cohen
16:00-16:30  - Break / coffee / breakout rooms
16:30-16:50  - Simon Baker
17:00-17:20  - Sascha Trocheit
17:30-17:50  - Jimmy Tseng

Monday June 29
13:30-14:00  - Zoom room opens
14:00-14:50  - Tanja Eisner
15:00-15:20  - Tanja Schindler
15:30-15:50  - Dominik Kwietniak
16:00-16:30  - Break / coffee / breakout rooms
16:30-16:50  - Nikolai Edeko
17:00-17:20  - Henrik Kreidler
17:30-17:50  - Cagri Sert

We plan to record the talks and make the recordings available on the workshop website (https://ergwork.web.unc.edu/schedule-of-talks-201/). If you don’t want your talk to be recorded, let us know. After each block of talks, we will open breakout rooms, one for each speaker, in order for the participants to interact with the speakers. We will conclude each of the two days with an informal coffee break / open discussion. This will also be an opportunity to discuss open questions. The breakout room conversations and the open discussions will not be recorded.

Best regards,

Zoltán Buczolich
Natalia Jurga
Paulo Varandas
(on behalf of the organizing committee)

A conference on Dynamical Systems
Homepage: http://dyngroups.math.uni.lodz.pl/

Faculty of Mathematics and Computer Science, University of Lodz, June 22 - 25, 2021.

The organizers of the international conference "Dynamics of (Semi-)Group Actions" warmly welcome all mathematicians interested in dynamics of (semi-)groups. This would be the second edition, the first one took place in July 2019 in Łódź (Poland). This time, due to widely known circumstances it will be virtual conference only.

The first aim of the conference is to bring together mathematicians involved in topological, measurable and algebraic aspects of dynamical systems generated by (semi-)group actions. The second aim is to explore research advances and to stimulate the exchange of new ideas. In addition, the conference offers new opportunities for broadening scientific contacts.

Lectures start on Tuesday (June 22) morning and end on Friday (June 25) afternoon.

Main speakers

• Vitaly Bergelson (Ohio State University, USA)

• Raimundo Briceno (PUC - Santiago de Chile)

• Anna Giordano Bruno (University of Udine, Italy)

• Alexander Bufetov (Aix-Marseille Universite, CNRS, France; Steklov Mathematical Institute of RAS, Russia)

• Maria Carvalho (University of Porto, Portugal)

• Eli Glasner (Tel-Aviv University, Israel)

• Vadim Kaimanovich (University of Ottawa, Canada)

• Eugen Mihailescu (Romanian Academy, Romania)

• Mark Pollicott (University of Warwick, UK)

• Feliks Przytycki (Polish Academy of Science, Poland)

• Michał Rams (Polish Academy of Science, Poland)

• Manuel Stadlbauer (Federal University of Rio de Janeiro, Brazil)

• Hiroki Sumi (Kyoto University, Japan)

• Tom Ward (University of Leeds, UK)

• Benjamin Weiss (Hebrew University of Jerusalem, Israel)

Date. June 18, 14h00 (UTC/GMT+1)

Speaker.  Vilton PINHEIRO (Federal University of Bahia)

Title. Ergodic formalism and topological attractors

Abstract.

We introduce the concept of Baire Ergodicity and Ergodic Formalism. Using them to study topological and statistical attractors, we give several applications for maps of the interval, expanding and partially hyperbolic maps, strongly dissipative dynamics and skew-products.  In dynamical systems with abundance of historical behavior (and this includes all systems with some hyperbolicity, in particular, Axiom A systems), we cannot use an invariant probability to control the asymptotic topological/statistical behavior of a generic orbit, except when it belongs to the basin of a periodic orbit of attraction. However, Baire's Ergodicity and Ergodic Formalism can also be applied in this context, contributing to the study of generic orbits of systems with an abundance of historical behavior.

Date. June 11, 14h00 (UTC/GMT+1)

Speaker. Victoria SADOVSKAYA (Penn State University)

Title. Diffeomorphism cocycles over partially hyperbolic systems

Abstract.

We consider cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold M. We will discuss the following results for this setting: existence of a continuous invariant family of Riemannian metrics on M for a bounded cocycle, continuity of a measurable conjugacy between two cocycles, and a sufficient condition for existence of a continuous conjugacy. 

Date. June 04, 14h00 (UTC/GMT+1)

Speaker.  Dalia TERHESIU (Leiden University)

Title. Lorentz gases with small scatterers

Abstract.

We study Lorentz gases with scatterer size  going to zero. In work in progress with Balint and Bruin, we obtain a precise understanding of the allowed path of taking limits as the size of the scatterers goes to zero and time goes to infinity so that the Central Limit Law (with non-standard normalization) persists. This type of joint law is in the gist of Boltzmann–Grad limit. Further we obtain joint local limit theorems and mixing as the size of the scatterers goes to zero and time goes to infinity.

Date. May 28, 16h00 (UTC/GMT+1) 

Speaker.  Pablo SHMERKIN (University of British Columbia)

Title. Beyond Furstenberg's intersection conjecture

Abstract.

Hillel Furstenberg conjectured in the 1960s that the intersections of closed ×2 and ×3-invariant Cantor sets have “small” Hausdorff dimension. This conjecture was proved independently by Meng Wu and by myself; recently, Tim Austin found a simple proof. I will present some generalizations of the intersection conjecture and other related results.

Date. May 21, 14h00m (UTC/GMT+1)

Speaker.  Tushar DAS (University of Wisconsin - La Crosse)

Title. Thermodynamic formalism for coarse expanding dynamical systems

Abstract.

We report on a joint project (https://arxiv.org/abs/1908.08270 or https://doi.org/10.1007/s00220-021-04058-2) with Giulio Tiozzo, Feliks Przytycki, Mariusz Urbanski, and Anna Zdunik that develops the thermodynamic formalism for weakly coarse expanding dynamical systems (finite branched covers of locally connected spaces that possess some expansion). There are no assumptions of finiteness of the postcritical set, conformality, holomorphy, or smoothness in this setting. After encoding the dynamics via geometric code trees, our stochastic laws follow from those for the shift dynamics on the associated code space. The area is in its nascency and there are plenty of open problems and directions that await exploration.

Date. May 14, 14h00m (UTC/GMT+1)

Speaker.  Polina VYTNOVA (Warwick University)

Title. Computing Hausdorff dimension of Bernoulli convolutions

Abstract.

It is well known that for any 0<s<1 the sum of the infinite geometric series 1+s+s^2+ .... is 1/(1-s). Let us consider a similar expression where the signs are chosen at random, i.e. "+" and "-" in front of   every term s^n are chosen with equal probability 1/2. The sum of the infinite series with randomly chosen signs is a random variable, and its law is a probability measure called the Bernoulli convolution.  

It turns out the properties of this measure are very sensitive to the choice of s. For instance, if s<1/2, then it is supported on the Cantor set; if s = 1/2 then it is equal to the Lebesgue measure on [-2,2]. The case of s>1/2 is very intricate and has been a subject of intensive research which dates back to Erdos. To date, it is not known whether or not the measure is singular or absolutely continuous for a given s. The problem of computing the Hausdorff dimension of the measure is also open in general, though there have been several major advances in recent years. 

 In the talk I will discuss a new approach to uniform lower bounds on Hausdorff dimension of Bernoulli convolution measures, based on random dynamics.

Online Zoom meeting (Session will open some minutes before 14h00):
https://videoconf-colibri.zoom.us/j/81658357378?pwd=MU50UVF3K09GcEFnQWpSK0V4ZE9nUT09

Meeting ID: 816 5835 7378

Password: 183424

Date. May 07, 14h00m (UTC/GMT+1)

Speaker. Sergey KRYZHEVICH (Saint Petersburg State University)

Title. Interval translation maps, their invariant measures and their properties

Abstract.

We study properties of the so-called interval translation maps (ITMs). Informally, we consider interval exchange maps with overlaps. A survey on the recent developments in this area will be given with some author’s results being mentioned. For example, we prove that any ITM admits a Borel probability non-atomic invariant measure. We introduce the so-called maximal invariant measure that is the measure supported on the largest possible set. It will be demonstrated that any ITM, endowed with this measure, is isomorphic to an interval exchange map.    This will imply various corollaries: the estimate on the maximal number of non-atomic ergodic invariant measures, minimality if the symbolic model, unique ergodicity, ‘closing lemma’, etc. Besides, we discuss the problem of robustness of invariant measures with respect to small perturbations of the parameters of the system.  Finally, a real-life model of risk management will be considered. We demonstrate that this model can be reduced to an ITM and study the properties of the obtained map.

Date. April 30, 14h00m (UTC/GMT+1)

Speaker.  Vaughn CLIMENHAGA (University of Houston) 

Title. Counting closed geodesics on surfaces without conjugate points

Abstract.

For negatively curved Riemannian manifolds, Margulis gave an asymptotic formula for the number of closed geodesics with length below a given threshold. I will describe joint work with Gerhard Knieper and Khadim War in which we obtain the corresponding result for surfaces without conjugate points by first proving uniqueness of the measure of maximal entropy and then following the approach of recent work by Russell Ricks, who established the asymptotic estimates in the setting of CAT(0) geodesic flows.

Date. April 23, 14h00m (UTC/GMT+1)

Speaker. Piotr OPROCHA (AGH, University of Science and Technology - Krakow)

Title. On dynamics of Lorenz maps

Abstract.

In this talk we will present several dynamical properties of expanding Lorenz maps, which roughly speaking are increasing interval maps with one point of discontinuity. We will put special emphasis on piecewise linear case:  x -> beta x + alpha (mod 1)  Among other, we will consider such properties as transitivity, mixing, renormalizations, minimal cycles and proper invariant sets.

Date. April 16, 10h00 (UTC/GMT+1) 

Speaker. Gary FROYLAND (University of New South Wales)

Title. Stability and approximation of statistical limit laws     

Abstract.

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. We prove stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We then apply these results to piecewise expanding maps in one and multiple dimensions. This theory can be extended to uniformly hyperbolic maps and in this setting we develop two new Fourier-analytic methods to provide the first rigorous estimates of the variance and rate function for Anosov maps.  This is joint work with Harry Crimmins.

Date. April 09, 14h00m (UTC/GMT+1)

Speaker. Maria CARVALHO (University of Porto) 

Title. Generalized entropy functions, variational principles and equilibrium states 

Abstract.

In this talk, based on joint work with Andrzej Bis´, Miguel Mendes and Paulo Varandas, I will present a conceptual approach to the thermodynamic formalism of dynamical systems and semigroup actions. More precisely, using methods from Convex Analysis, I will show how to construct for each pressure function an upper semi-continuous affine entropy-like map (which, in the context of continuous transformations acting on a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy). This strategy also provides an abstract variational principle and guarantees that equilibrium states always exist, though they may be at best  finitely additive. Afterwards, I will discuss some applications in a wide range of topics.

Date. March 26, 14h00m (UTC/GMT)

Speaker. Bálasz BÁRÁNY (Budapest University of Technology and Economics)

Title. On the convergence rate of the chaos game

Abstract.

 In the 1988 textbook "Fractals Everywhere" M. Barnsley introduced an algorithm for generating fractals through a random procedure which he called the chaos game. In this talk we show how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. As an application, for Bedford-McMullen carpets we completely characterise the family of probability vectors which minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford-McMullen carpets before. This is a joint work with Natalia Jurga and István Kolossváry.

Date. March 19, 14h00m (UTC/GMT)

Speaker.  Yuri LIMA (Universidade Federal do Ceará)

Title. Symbolic dynamics for maps with high dimension

Abstract. 

We construct Markov partitions for non-invertible and/or singular nonuniformy  hyperbolic systems defined on higher dimensional Riemannian manifolds. The gen- erality of the setup covers classical examples not treated so far, such as geodesic  flows in closed manifolds, multidimensional billiard maps, and Viana maps, as well as includes all the recent results of the literature. We also provide a wealth of applications. Joint work with Ermerson Araujo and Mauricio Poletti.

Date. March 12, 14h00m (UTC/GMT)

Speaker.  Jerôme BUZZI (Faculté des Sciences d'Orsay, Paris-Saclay)

Title. Continuity of Lyapunov exponents for surface diffeomorphisms

Abstract.

In the setting of smooth surface diffeomorphisms, we show that the lower semicontinuity defect of the exponents bounds that of the entropy. In particular, Hausdorff dimension is upper semicontinuous among ergodic invariant probability measures with entropy bounded away from zero. We also obtain a new criterion for the existence of an SRB measure with positive entropy. Joint work with Sylvain CROVISIER and Omri SARIG.

Date. March 05, 13h00 (UTC/GMT) 

Speaker.  Mao SHINODA (Ochanomizu University)

Title. Non-convergence of equilibrium states at zero temperature limit

Abstract.

We construct finite-range interactions on a two dimensional symbolic dynamical system, for which the associated equilibrium states (i.e., the shift-invariant Gibbs states) fail to converge as temperature goes to zero. More precisely, if we pick any one-parameter family of equilibrium states at inverse temperature for this interaction, then the limit does not exist. This settles a question posed by Chazottes and Hochman who obtained such a non-convergence behavior when dimension greater than 3.

Date. February 26, 14h00m (UTC/GMT)

Speaker.  Federico RODRIGUEZ-HERTZ (Penn State University)

Title. Rigidity for expanding maps

Abstract.

In this talk I will discuss a plan started joint with A. Gogolev to find rigidity property in hyperbolic dynamics. This plan focuses on matching of smooth potentials for different dynamics. I will present a successful instance of this plan in the setting of expanding maps.

Date. February 19, 14h00m (UTC/GMT)

Speaker.  Benoît KLOECKNER (Université Paris-Est)

Title. Effective high-temperature estimates ensuring a spectral gap

Abstract.

The main goal of the talk shall be to explain a few ideas from two classical theories : the thermodynamical formalism, and the perturbation of linear operators. The "thermodynamical formalism" is a framework to describe particular invariant measures of dynamical systems, called "equilibrium states", parametrized by functions on the phase space, called "potentials". This formalism is based on the "transfer operator"; when this operator has a spectral gap, the equilibrium state exists, is unique, and has very good statistical properties (exponential mixing, Central Limit Theorem, etc.) If one perturbs slightly the potential, the corresponding transfer operator is also perturbed. The classical theory of perturbation of operators ensures that the spectral gap property is an open condition and that under bounded pertubration, the eigendata of an operator depends analytically on the perturbation. It turns out that using the Implicit Function Theorem, this theory can be made effective with explicit bounds on the size of a neighborhood where the spectral gap persists. Using this effective perturbation theory, we show completely explicit bound on the potential ensuring the spectral gap property for transfer operators of classical families of dynamical systems.

Date. February 12, 14h00m (UTC/GMT)

Speaker.  Natalia JURGA (University of St. Andrews)

Title. Box dimensions of $(\times m, \times n)$ invariant sets

Abstract.

We study the box dimensions of sets which are invariant under the toral endormorphism $(x, y) \mapsto (m x \mod  1, n y  mod 1)$ for integers $n>m \geq 2$. This is a fundamental example of an expanding, nonconformal dynamical system, and invariant sets have many subtle properties. The basic examples of such invariant sets are Bedford-McMullen carpets and, more generally, invariant sets are modelled by subshifts on the associated symbolic space. When this subshift is topologically mixing and sofic, the situation is well-understood by results of Kenyon and Peres, in particular the box dimension satisfies a natural formula in terms of entropy and the expansion coefficients $m,n$.  In this talk we will discuss what happens beyond the sofic and mixing case, which is partly based on joint work with Jonathan Fraser (St Andrews).

Date. February 05, 14h00m (UTC/GMT)

Speaker.  Eugen Mihailescu (Institute of Mathematics of the Romanian Academy)

Title. Dynamics for a class of skew-product endomorphisms and applications

Abstract.

I will present a new notion of Smale endomorphisms and the skew-products coded by them. Equilibrium measures are studied for these piecewise smooth transformations; their projections are shown to be exact dimensional in fibers and globally, and dimension formulas are obtained. This applies to skew-products over EMR maps and natural extensions. Then we extend the Doeblin-Lenstra Conjecture on Diophantine approximation coefficients for a larger class of measures and of irrational numbers. Another application is to a type of complex continued fractions with complex digits. Based on joint work with M. Urbanski.

Date. January 29, 14h00m (UTC/GMT)

Speaker.  Ana Cristina FREITAS (Universidade do Porto)

Title. The cluster size distribution and the extremal index

Abstract.

The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits. In this context, the existence of an extremal index less than 1 is associated to the occurrence of periodic phenomena. For generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The extremal index usually coincides with the reciprocal of the mean of the limiting cluster size distribution. Here, we build dynamically generated stochastic processes with an extremal index for which that equality does not hold. The mechanism we use is based on considering observable functions maximized at at least two points of the phase space, where one of them is an indifferent periodic point. For the second point at which the observable function is maximized we consider either a periodic or a nonperiodic point. We explore two such examples and, for each of them, we compute the extremal index and present the corresponding cluster size distribution.

Date. January 22, 14h00m (UTC/GMT)

Speaker.  Jairo Bochi (PUC Chile)

Title. Topological emergence

Abstract.

I will explain the notion of topological emergence of a dynamical system, introduced by Pierre Berger and myself (2019). Topological emergence is a measurement of diversity of statistical behavior, and it is independent of the traditional notions of dynamical complexity, such as entropy. I will show an universal bound on topological emergence depending on the ambient dimension, and I will discuss the problem of computing topological emergence of some hyperbolic dynamical systems. I will sketch the proofs.

Date. January 15, 14h00m (UTC/GMT)

Speaker.  Hong-Kun ZHANG (University of Massachusetts Amherst)

Title. Levy diffusion for Lorentz gas with flat points

Abstract. 

We  investigate the diffusion and statistical  properties of  Lorentz gas with cusps at flat points. The underlying dynamical systems  are modifications of  dispersing billiards  with cusps. The  systems  generate stochastic processes with slow mixing rates of order  $n^{-a$}, for $ a\in (0,1)$. We are able to show  that these  stochastic processes   enjoy stable law and have super-diffusion driven by   Levy processes.

Date. December 18, 14h00m (UTC/GMT)

Speaker.  Giulio Tiozzo (University of Toronto)

Title. Central limit theorems for counting measures in coarse negative curvature

Abstract.

We establish general central limit theorems for an action of a group on a hyperbolic space with respect to counting for the word length in the group. In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the corresponding closed geodesic on the pair of pants.  Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem, and we proved this conjecture in 2018. In our new work, we remove the assumptions of properness and smoothness of the space, or cocompactness of the action, thus proving a general central limit theorem for group actions on hyperbolic spaces. We will see how our techniques replace the classical thermodynamic formalism and allow us to provide new applications, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds. Joint work with I. Gekhtman and S. Taylor.

Date. December 11, 14h00m (UTC/GMT)

Speaker.  Andrés Koropecki (Universidade Federal Fluminense)

Title. A model factor map for surface diffeomorphisms

Abstract.

We show that if a $C^{1+\alpha}$ diffeomorphism $f$ of $\mathbb{T}^2$ homotopic to the identity has a rotation set with nonempty interior, then it is monotonically semiconjugate to a homeomorphism $F$ of $\mathbb{T}^2$ which is area-preserving, topologically mixing, has dense periodic points and every point has a nontrivial stable and unstable set. Moreover, it has a strong form of continuum-wise expansiveness. Further, $F$ has the same rotation set as $f$, so one consequence is that every rotation set realizable by a $C^{1+\alpha}$ diffeomorphism is also realizable by an area-preserving homeomorphism with all these properties. We also obtain a similar result on surfaces of higher genus, with the condition on the rotation set replaced by the existence of certain types of periodic orbits. Joint work with A. de Carvalho and F. A. Tal.

Date. December 04, 14h00m (UTC/GMT)

Speaker.  Jaqueline Siqueira (Universidade Federal do Rio de Janeiro)

Title. Equilibrium states for a class of non-uniformly hyperbolic maps

Abstract.

We consider a wide family of non-uniformly hyperbolic maps and hyperbolic potentials and prove that the unique equilibrium state associated to each element of the family is given by the eigenmeasure and the eigenfunction of the transfer operator (both having the spectral radius as an eigenvalue). We prove that the transfer operator has the spectral gap property in the space of Hölder continuous observables. From this we derive that the unique equilibrium state satisfies a central limit theorem and that it has exponential decay of correlations. Moreover, we prove joint continuity and analyticity with respect to the potential. (Based on various joint works with S. Afonso, J. Alves, V. Ramos.)

Date. November 27, 14h00m (UTC/GMT)

Speaker.  Carlos Matheus (Centre de Mathématiques Laurent Schwartz)

Title. Speed of mixing of geodesic flows on certain non-positively curved surfaces

Abstract.

In this talk, we discuss the speed of mixing on certain surfaces whose curvatures are negative except along a closed geodesic. This is a joint work in progress with Y. Lima and I. Melbourne

Date. November 20, 14h00m (UTC/GMT)

Speaker.  Yeor Hafouta (Ohio State University)

Title. Limit theorems for some time-dependent expanding dynamical systems

Abstract.

We will discuss various probabilistic limit theorems for some classes of distance expanding time-dependent dynamical systems. Some of the results like the Berry-Esseen theorem and moderate deviations principle hold true for general sequences of maps when the variance of the underlying partial sums grows faster than n^{2/3}, while other results such  the local central limit theorem hold true for certain classes of random not necessarily stationary transformations.  The results also include a certain type of stability theorem in a complex version of the sequential Ruelle-Perron-Frobenius theorem, which yields that the variance grows linearly fast when the underlying maps are close enough to a single expanding map.

Date. November 13, 14h00m (UTC/GMT)

Speaker.  Romain Aimino (Porto)

Title. Large deviations estimates for dynamical systems with stretched exponential decay of correlations

Abstract.

During this talk, I will discuss results on large deviations estimates for dynamical systems with stretched exponential decay of correlations. This class of systems encompasses for instance Viana maps and intermittent maps exhibiting a stretched exponential fast accumulation of pre-orbits at 0. Optimal rates for large deviations of such systems are still unknown. I will present a substancial improvement obtained in a joint work with Jorge Freitas.

Date. November 06, 14h00m (UTC/GMT)

Speaker.  Christian Wolf (City University of New York)

Title. Computability of topological pressure on compact shift spaces beyond finite type​

Abstract.

In this talk we discuss the computability (in the sense of computable analysis) of the topological pressure $P_{\rm top}(\phi)$ on compact shift spaces $X$ for continuous potentials $\phi:X\to\bR$. This question has recently been  studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and  apply this framework to  coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts,  generalized gap shifts, and Beta shifts. We also construct shift spaces which, depending on the potential,  exhibit  computability and non-computability of the topological pressure.  We further show  that the generalized pressure function $(X,\phi)\mapsto P_{\rm top}(X,\phi\vert_{X})$ is not computable for a large set of shift spaces $X$ and potentials $\phi$.  Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability. The topic of the talk is joint work with Michael Burr (Clemson U.), Shuddho Das (NYU) and Yun Yang (Virginia Tech).

Date. October 30, 14h00m (UTC/GMT)

Speaker.  Yiwei Zhang (Huazhong University of Science and Technology)

Title. Structure of optimal orbits for chaotic dynamical systems: sophisticated vs simple

Abstract. 

Given a topological dynamical system  $T:X\to X$, and an continuous observable $\varphi:X\to\mathbb{R}$, we say an orbit $\mathcal{O}_{x_{0}}=\{x_{0},T(x_{0}),\cdots\}$ is an $f$-optimal orbit, if the Birkhoff average $\langle \varphi\rangle(x_{0}):=\lim_{n\to\infty}\frac{1}{n}\varphi(T^{i}(x_{0}))$ exists, and $\langle\varphi\rangle(x_{0})\geq\limsup_{n\to\infty}\frac{1}{n}\varphi(T^{i}(x)),\forall x\in X$. Define by $\mathcal{S}_{op}\subset X$, the set of all initial states, which give rise to the optimal orbit.  We will investigate the geometric structure of $\mathcal{S}_{op}$, and see how the structure of $\mathcal{S}_{op}$ varies, corresponding to the different level of hyperbolicity of $T$, and regularity of $\varphi$.

Date. October 23, 14h00m (UTC/GMT+1)

Speaker.  Stefano Galatolo (Università di Pisa)

Title. Linear response and control of the statistical properties of dynamics

Abstract. 

We will review the concept of linear response with emphasis to the case of dynamical systems with additive noise. This is an example of a quite general class of random systems for which one has linear response results without assuming hyperbolicity conditions on the deterministic part of the dynamics, thank to the regularizing effect of the noise. We will then discuss the control problem naturally associated to linear response: what is the best perturbation to be applied to a system in order to get some wanted change in its statistical properties? This problem has evident importance in the applications. We briefly discuss the mathematical structure of the problem, showing efficient methods for the approximation of the optimal solutions and show some examples.

Date. October 16, 14h00m (UTC/GMT+1)

Speaker. Yushi Nakano (Tokai University)

Title. Large intersection classes for pointwise emergence

Abstract.

In this talk, I introduce a concept of pointwise emergence to quantitatively study sets without time averages (called irregular sets, or sets with historic behavior), which is inspired by a recent work by P. Berger. We show that for any topologically mixing subshift of finite type, there exists a residual subset of the state space with high pointwise emergence, full topological entropy, full Hausdorff dimension, and full topological pressure for any H older continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property. This is a natural generalization of [Farm-Persson2011] to pointwise emergence and Caratheodory dimension. This is a joint work with  A. Zerelowicz.

The Symposium in Dynamical Systems, hosted by University of Porto, on the 14th of October 2020 is near. The meeting, on the occasion of Jorge Rocha 60th birthday, will focus on recent progress in Dynamical Systems, with particular emphasis on Lyapunov exponents, entropy, centralizers and heterodimensional cycles. Here goes some updated information.

List of Speakers:
José F. Alves (Universidade do Porto)
Lorenzo Díaz (PUC-Rio de Janeiro)
Pedro Duarte (Universidade de Lisboa)
Davi Obata (Chicago University)
Maria J. Torres (Universidade do Minho)
Marcelo Viana (IMPA)

Date. October 09, 14h00m (UTC/GMT+1)

Speaker.  Kiho Park (University of Chicago)

Title. Transfer operators and limit laws for typical cocycles

Abstract.

We show that typical cocycles (in the sense of Bonatti and Viana) over irreducible subshifts of finite type obey several limit laws with respect to the unique equilibrium states for H\”older potentials. These include the central limit theorem and the large deviation principle. We also establish the analytic dependence of the top Lyapunov exponent on the underlying equilibrium state. The transfer operator and its spectral properties play key roles in establishing these limit laws.

Date. October 02, 14h00m (UTC/GMT+1)

Speaker.  Leandro Cioletti(Universidade de Brasília)

Title. The space of harmonic functions for transfer operators and phase transitions

Abstract.

It was recently proved that the set of DLR-Gibbs measures, associated with a uniformly absolutely summable interaction on the lattice N, coincides with the set of P(f)-conformal measures associated with a suitable continuous potential f. In this talk, we explore this result and prove a new relation between the set of extreme P(f)-conformal measures and the dimension of the Perron-Frobenius eigenspace of the L1-extension of the transfer operator associated with the potential f. In particular, we show that such eigenspace's geometric multiplicity can only be greater than one when a first-order phase transition occurs. We obtain this result by looking at the transfer operator's extension as a Markov Process in the Hopf's sense. Applications for equilibrium states associated with low regular potentials will be discussed, and an interesting example, where the space of harmonic functions has dimension two, will be presented. We will finish with a discussion on the Functional Central Limit Theorem for equilibrium states and non-local observables, which holds for observables in the Banach space of generalized Hölder continuous functions, where the transfer operator acts without the spectral gap property.

Date. September 25, 14h00m (UTC/GMT+1)

Speaker. Jana Rodriguez-Hertz (SUSTech)

Title. Robust minimality of strong foliations for DA diffeomorphisms:  new examples

Abstract.

Let f be a C^2 partially hyperbolic diffeomorphisms of T^3 (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism A with eigenvalues k_3<1<k_2<k_1. If the set \{x: \mid \log \det(Tf\mid_{E^{cu}(f)})\mid \leq \log k_1 \} has zero volume inside any unstable leaf of f, then the stable foliation of f is C^1 robustly minimal, i.e., the stable foliation of any diffeomorphism C^1 sufficiently close to f is minimal. In particular, f itself is robustly transitive.
We build, with this criterion, a new example of a C^1 open set of DA diffeomorphisms, such that the strong stable foliation and the strong unstable foliation of any diffeomorphism in this open set are both minimal. The existence of such an example was unknown in this setting. This is a joint work with  R. Ures and J. Yang.

Date. September 18, 14h00m (UTC/GMT+1)

Speaker. Pierre-Antoine Guiheneuf (Université Paris Sorbonne)

Title. Historic behaviour vs. physical measures for irrational flows with multiple stopping points

Abstract.

When studying the ergodic properties of a dynamical system, one can look at the dichotomy convergence vs. divergence of Birkhoff averages for a positive Lebesgue measure set of points. In this talk, we will study this question in the case of a linear irrational flow of the 2 torus reparametrized at two stopping points. In particular, we will see that the answer depends on the Diophantine type of the slope of the flow.

Date. September 04, 13h00m UNSUAL TIME (UTC/GMT+1)

Speaker. Katrin Gelfert (Federal University of Rio de Janeiro

Title. Skew-products with concave fiber maps

Abstract.

We consider skew-products with concave interval fiber maps over a certain subshift. Here the subshift occurs as the projection of those orbits that stay in a given neighborhood and gives rise to a new type of symbolic space which is (essentially) coded. The fiber maps have expanding and contracting regions. As a consequence, the skew-product dynamics has pairs of horseshoes of different type of hyperbolicity. In some cases, they dynamically interact due to the superimposed effects of the (fiber) contraction and expansion, leading to nonhyperbolic dynamics that is reflected on the ergodic level (existence of nonhyperbolic ergodic measures).

We provide a description of the space of ergodic measures on the base as an entropy-dense Poulsen simplex. Those measures lift canonically to ergodic measures for the skew-product. We show how that these skew-products can be embedded in increasing entropy one-parameter family of diffeomorphisms which stretch from a heterodimensional cycle to a collision of homoclinic classes. We study associated bifurcation phenomena that involve a jump of the space of ergodic measures and, in some cases, also of entropy. (Joint work with L.J.Díaz and M.Rams)

Date. July 31, 14h00m (UTC/GMT+1)

Speaker. Yi Shi (Peking University)

Title. The space of ergodic measures for Lorenz attractors

Abstract.

In this talk, we show that the space of ergodic measures is path connected for C^r(r>1)-generic Lorenz attractors while it is not connected for C^r-dense Lorenz attractors. Various properties of the ergodic measure space for Lorenz attractors have been showed. In particular, a C^r(r>1)-connecting lemma for Lorenz attractors also has been proved. This is a joint work with Xueting Tian and Xiaodong Wang.

Date. July 24, 14h00m (UTC/GMT+1)

Speaker. Rodrigo Bissacot (Universidade de São Paulo)

Title. Thermodynamic Formalism: from Generalized to Standard Countable Markov Shifts and Conservative-Type Phase Transitions

Abstract.

Given a transitive 0-1 infinite matrix $A$, we develop the thermodynamic formalism over a {\it generalized countable Markov shift} $X_A$. This locally compact space contains as dense subsets, the usual countable Markov $\Sigma_A$, and a collection of allowed words (respect to $A$), which we call $Y_A$. Defined by M. Laca and R. Exel in 1999, the object was introduced when they studied the corresponding operator algebras as a generalization of the Cuntz-Krieger algebras for an infinite and countable alphabet. The differences are that shift map $\sigma$ is partially defined and, $X_A$ is always locally compact since is the spectrum certain C* - algebra (weak* topology).

We introduce the notion of conformal measures in $X_A$ and, exploring the connection with the usual thermodynamic formalism in $\Sigma_A$, new phenomena appear:

- Different types of phase transitions in $\Sigma_A$. (conservative-type)

- New conformal measures that are not detected in classical thermodynamic formalism.

Given a potential $f$, we study the problem of the existence and absence of conformal measures $\mu_{\beta}$ associated with $\beta f$ living on $X_A$ for different values of $\beta > 0$. We present an example where there exists a critical $\beta_c$ such that we have the existence of conformal probability measures satisfy $\mu_{\beta}(\Sigma_A)=0$ for every $\beta > \beta_c$ and, when we take the limit $\beta$ going to $\beta_c$, $\mu_{\beta}$ converges to $\mu_{\beta_c}$ and $\mu_{\beta_c}(\Sigma_A)=1$.

Combining results from Sarig and Denker-Yuri approaches, and using the fact $X_A$ is locally compact, we highlight a new type of phase transition in countable Markov shifts $\Sigma_A$: a {\it conservative-type phase transition}. We exhibit a example of a potential $f$ which admits a conformal measure $\mu_{\beta}$ associate to $\beta f$ for every $\beta >0$ but, in this particular example, there exists a critical $\beta^{'}_c$ such that $\mu_{\beta}$ is conservative for $\beta < \beta^{'}_c$ and not conservative for $\beta > \beta^{'}_c$.

Jointly work with Thiago Raszeja (USP), Ruy Exel - Federal University of Santa Catarina (UFSC), and Rodrigo Frausino - University of Wollongong (UOW).

References:

[BEFR1] R. Bissacot, R. Exel, R. Frausino and T. Raszeja, Conformal measures on generalized RenaultDeaconu groupoids, arXiv:1808.00765.

[BEFR2] R. Bissacot, R. Exel, R. Frausino and T. Raszeja, Quasi-invariant measures for generalized approximately proper equivalence relations, arXiv:1809.02461.

[DY] M. Denker and M. Yuri, Conformal families of measures for general iterated function systems. Contemporary Mathematics, 631, 93-108, (2015).

[EL] R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices. J. Reine Angew. Math., 512, 119-172, (1999).

[Sa]O. Sarig. Thermodynamic formalism for countable Markov shifts. Proc. of Symposia in Pure Math. \textbf{89} (2015), 81-117

Date. July 17, 14h00m (UTC/GMT+1)

Speaker. Jiagang Yang (Universidade Federal Fluminense)

Title. Recent progress on partially hyperbolic diffeomorphisms with mostly expanding centers

Abstract.

Partially hyperbolic diffeomorphisms with mostly expanding centers were first introduced by Alves, Bonatti, and Viana as a generalization of the derived from Anosov diffeomorphisms by Mane. They showed that such systems always admit physical measures. It was further proven by Alves and Li, under extra hypotheses on the tail of the hyperbolic times, that those physical measures have exponential decay of correlations for Holder observables.

In this talk, we will introduce the recent progress on partially hyperbolic diffeomorphisms with mostly expanding centers, including Anderson and Vasquez's new definition using Gibbs $u$-states; the perturbation theory for physical measures; skeletons; and how to use physical measures to prove topological transitivity. New examples will also be provided.

Date. July 10, 14h00m (UTC/GMT+1)

Speaker. Wael Bahsoun (Loughborough University):

Title. Anisotropic BV spaces

Abstract.

Given any smooth Anosov map we construct a Banach space on which the associate transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. This is a joint work with Carlangelo Liverani.

Date. July 03, 14h00m (UTC/GMT+1)

Speaker. Alexey Korepanov (University of Exeter)

Title. Concentration inequalities in non-stationary chaotic deterministic dynamical systems

Abstract.

For a random process, concentration inequalities give bounds for probability of deviations far from average. Well known examples of concentration inequalities are moment bounds, large and moderate deviations. These are classic for sufficiently chaotic processes, in particular for (stationary) deterministic dynamical systems. I will talk about concentration inequalities for non-stationary dynamics, designed to model a changing environment, where much less is known. This is a joint work with Juho Leppänen

Date. June 26, 14h00 (UTC/GMT+1)

Speaker. Hélder Vilarinho (Universidade da Beira Interior)

Title. Liftability of expanding stationary measures

Abstract.

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold and discuss the lift of absolutely continuous ergodic stationary measures which are expanding (all Lyapunov exponents positive) via random Gibbs-Markov-Young structures.

We also concern the stability of expanding measures: we will see that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding. This is a joint work with José F. Alves and Carla L. Dias.

Date. June 19, 13h30 (UTC/GMT+1)

Speaker. Xueting Tian (Fudan University, China)

Title. Research on Points without Lyapunov exponents

Abstract.

The It follows from Oseledec Multiplicative Ergodic Theorem (or Kingman’s sub-additive Ergodic Theorem) that the set of ‘non-typical’ points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any Holder continuous cocycles over hyperbolic systems, in this talk we show that either all ergodic measures have same Maximal Lyapunov exponents or the set of Lyapunov ‘non-typical’ points is a dense $G_\delta$ subset and carries full topological entropy and packing topological entropy. Moreover, we give an estimate of Bowen Hausdorff entropy from below by the metric entropy of ergodic measures which are not Lyapunov minimizing, and if further the function of integrable Lyapunov exponent is lower semi-continuous with respect to invariant measures, the set of Lyapunov ‘non-typical’ points carries full Bowen Hausdorff entropy.

Date. June 12, 14h00 (UTC/GMT+1)

Speaker. Manuel Stadlbauer (Universidade Federal do Rio de Janeiro)

Title. Quenched and annealed decay of correlations, and a boundary of equilibria

Abstract.

The joint action of a finite number of Ruelle expanding maps might be studied by means of thermodynamic formalism as this approach, among other tools, comes with a natural selection principle through the variational principle. In order to do so, we discuss the asymptotics of the operator semigroup generated by the Ruelle operators associated with Holder continuous potentials. That is, as a first step, we are interested in the asymptotic behaviour of sequences of individual transfer operators, leading to quenched equilibrium states.

In order to study the whole semigroup, we will discuss two approaches. The first is based on the asymptotic behaviour of the annealed operators, obtained as average over a probability with the Gibbs-Markov property. This gives rise to an annealed probability measure (though not invariant) which has exponential decay of correlations. Note that it is not necessary to require an independent noise, and the connection between measures is new. The second approach, up to the knowledge of the authors, is completely new: by identifying sequences in the semigroup which have asymptotically the same equilibrium, one obtains a compactification of the semigroup. This is joint work with Paulo Varandas and Xuan Zhang.

Date. June 05, 14h00 (UTC/GMT+1)

Speaker. Reza Mohammadpour (IMPAN-Warsaw)

Title. Lyapunov spectrum properties

Abstract.

In this talk we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type, and that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show both the continuity of the entropy spectrum at the boundary of Lyapunov spectrum for such cocycles, and the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition. We consider a subadditive potential $\Phi$. We obtain that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states of $t\Phi$ is a maximizing measure, and that the Lyapunov exponent and entropy of equilibrium states for $t\Phi$ converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures.

Date. May 22, 11h00 (UTC/GMT+1)

Speaker. Jason Atnip (University of New South Wales, Sydney)

Title. Thermodynamic Formalism for Random Weighted Covering Systems

Abstract.

In this talk we develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. We consider a general random contracting potential (in the sense of Liverani-Saussol-Vaienti) and we prove there exists a unique random conformal measure and a unique random equilibrium state for this potential. Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for the unique equilibrium state.

We will give several examples of our general theory. In particular, our results apply to random beta-transformations, random Gauss-Renyi maps, and random dynamics of non-uniformly expanding maps such as intermittent maps and maps with contracting branches.

Date. May 15, 14h00 (local time)

Speaker. Artur Oscar Lopes (Universidade Federal do Rio Grande do Sul)

Title. Ergodic transport

Abstract.

The classical Transport Theory (discrete time) is - basically - not a dynamical theory. We will present several recent results where there is an interplay between Ergodic Theory and Transport Theory. We will begin by recalling some basic classical results like Kantorovich duality and the slackness condition, and then we will state results in the so called Ergodic Transport Theory. For instance, we can ask: given a probability which is not invariant, how to characterize the invariant probability which is more close to this one under the Wasserstein metric? Another result claims that the dual of the Ruelle operator of a Holder potential is a contraction under the 1-Wasserstein metric when acting on the set of probabilities. One can also consider Thermodynamic Formalism for Gibbs plans.

Date. May 08, 14h00 (local time)

Speaker. Felipe García-Ramos (Universidad Autónoma de San Luis Potosí)

Title. The topological Markov property and homoclinic points of expansive algebraic actions

Abstract.

Joint work with Sebastián Barbieri and Hanfeng Li. We will introduce the topological Markov property for continuous group actions on compact metric spaces (a weakening of shadowing). With this we will show that an expansive algebraic action of an elementary amenable group has positive entropy if and only if it has a non-trivial homoclinic point. This approach generalizes algebraic results of Lind-Schmidt and Chung-Li and a topological result of Meyerovitch.