Seminar 10.31.19 Downarowicz

Title: Asymptotic Pairs Versus Positivity of Entropy

Speaker: Tomasz Downarowicz – Wroclaw University of Technology

Abstract: Consider a dynamical system (X,T) consisting of a compact metric space X and iterates of a self-homeomorphism T of this space. Topological entropy of the system depends on the speed of growth of complexity of orbits. Zero entropy means that this growth is subexponential and positive entropy corresponds to exponential growth. All this seems quite sophisticated and subtle. On the other hand, there is a very simple-minded notion of an asymptotic pair: two points x, y in X are asymptotic pair if their orbits come closer and closer together as time advances. It might seem surprising but this simple concept SUFFICES to distinguish between positive and zero entropy systems. During my talk I will try to familiarize the audience with the main ideas behind this result (which is due to Blanchard-Host-Ruette in one direction and D. and Lacroix in the other). Moreover, in recent time D., Oprocha and Zhang have obtained a similar criterion for positive entropy in actions of any countable abelian group. If time permits. I will briefly say why this case is very different from the Z-case.

Seminar 10.24.19 Lindsey

Title: Thurston’s Master Teapot

Speaker: Kathryn Lindsey – Boston College

Abstract: When a multimodal self-map of an interval is postcritically finite (PCF), its growth rate (the exponential of its topological entropy) is a special type of algebraic number called a weak Perron number. W. Thurston plotted the set of all Galois conjugates of growth rates of PCF unimodal maps; this visually stunning image revealed that this set has a rich and mysterious geometric structure. Thurston’s Master Teapot is a closely related 3D set. This talk will present some of the basic topological and geometrical properties of these sets. Based on joint work with C. Wu. H. Bray, D. Davis.

Seminar 10.17.19 Velozo

Title: Pressure in Symbolic Dynamics

Speaker: Anibal Velozo – Yale

Abstract: Symbolic dynamics is a particularly useful tool to understand smooth systems with certain hyperbolicity; this is typically done via a “coding”, i.e. a Markov partition/section or inducing schemes. Moreover, the thermodynamic formalism of finite state and countable Markov shifts is well understood, and many results can be pushed to the smooth setting via such codings. The plan of the talk is to overview some known results about the thermodynamic formalism of finite state and countable Markov shifts, and to discuss some recent works about the limiting behavior of the pressure of invariant measures in the non-compact setting. If time permits we will also discuss analogous results for flows.

Seminar 10.3.19 Siddiqi

Title: Decay of correlations for isometric extensions of Anosov flows

Speaker: Salman Siddiqi, Michigan

Abstract: I will briefly provide some historical context discussing known results on exponential correlation decay (or exponential mixing) for Anosov flows. I’ll summarize some classical techniques, and sketch a proof that locally accessible isometric extensions of Anosov flows are exponentially mixing under certain conditions – this includes, for example, some classes of frame flows and flows on principal bundles.

Seminar 9.12.19 War

Title: Open sets of exponentially mixing Anosov flows

Speaker: Khadim War, IMPA and University of Chicago

Abstract: We prove that an Anosov flow with C^1 stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and dim(E^s) = 1, dim(E^u) ≥ 2 then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This implies the existence of non-empty open sets of exponentially mixing Anosov flows. This is based on a joint work with Oliver Butterley.

Seminar 9.5.19 Butler

Title: Global rigidity of the periodic Lyapunov spectrum for geodesic flows of negatively curved locally symmetric spaces

Speaker: Clark Butler, Princeton

Abstract: We show that if a smooth Anosov flow f^{t} is orbit equivalent to the geodesic flow g^{t} of a negatively curved locally symmetric space X of dimension at least three and the Lyapunov spectra of the flow f^{t} at all periodic points are multiples of the corresponding Lyapunov spectra of g^{t} then f^{t} is smoothly orbit equivalent to g^{t}. If f^{t} is itself the geodesic flow of a negatively curved space Y then we further conclude that Y is homothetic to X. We deduce the Mostow rigidity theorem as a corollary.

Seminars for Fall 2019

Here is a list of visiting speakers currently scheduled for Fall semester 2019. All talks are on Thursdays in MW154 at 3.00pm-4.00pm (unless otherwise indicated). More talks will be announced soon.

Sept 5: Clark Butler (Princeton)

Sept 12: Khadim War (IMPA / Chicago)

Oct 3: Salman Saddiqi (Michigan)

Oct 17: Anibal Velozo (Yale)

Oct 24: Kathryn Lindsey (Boston College)

Oct 31: Tomasz Downarowicz (Wroclaw University, Poland)

Nov 7: Asaf Katz (Chicago)

Nov 14: Kurt Vinhage (Penn State)

Nov 21: Kiho Park (Chicago)

Seminar 4.18.19 Davis

Title: Tiling billiards and interval exchange transformations

SpeakerDiana Davis (Swarthmore College)

Abstract: Tiling billiards is a new dynamical system where a beam of light refracts through a planar tiling. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. I will explain this surprising correspondence, and will also discuss the behavior of the system for other interesting tilings.

Seminar 4.4.19 and 4.11.19 Gogolev

Title: What’s new in rigidity

SpeakerAndrey Gogolev (Ohio State University)

Abstract: A dynamical system is called rigid if a weak form of equivalence to a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. For Anosov dynamical systems smooth rigidity theory was initiated by Rafael de la Llave and collaborators who were motivated by the spectral rigidity program. We will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds.

In the first talk (April 4) I will explain what the problem is and explain some of the old results. Then I will state the new results. In particular, I will report on several improvements on Croke-Otal marked length spectrum rigidity.

In the second talk (April 11) I will present the proof ideas in their most basic form. Based on joint work in progress with Federico Rodriguez Hertz. I will make it accessible and explain all necessary background, everybody with interest in dynamics or geometry is welcome.

Seminar 3.28.19 Sadovskaya

Title: Periodic approximation of Lyapunov exponents for cocycles over hyperbolic systems

SpeakerVictoria Sadovskaya (Pennsylvania State University)

Abstract: We consider a hyperbolic dynamical system (X,f) and a Holder continuous cocycle A over (X,fwith values in GL(d,), or more generally in the group of invertible bounded linear operators on a Banach space. We discuss approximation of the Lyapunov exponents of A in terms of its periodic data, i.e. its return values along the periodic orbits of f. For a GL(d,)-valued cocycle A, its Lyapunov exponents with respect to any ergodic f-invariant measure can be approximated by its Lyapunov exponents at periodic orbits of f. In the infinite-dimensional case, the upper and lower Lyapunov exponents of A can be approximated in terms of the norms of the return values of A at periodic points of f. Similar results are obtained in the non-uniformly hyperbolic setting, i.e. for hyperbolic invariant measures. This is joint work with B. Kalinin.

Seminar 3.28.19 Chaika

Title: A prime transformation with many and big self-joinings

SpeakerJon Chaika (University of Utah)

Abstract: Let (X,μ,T) be a measure preserving system. A factor is a system (Y,ν,S) so that there exists F with SF=FT and so that F pushes μ forward to ν. A measurable dynamical system is prime if it has no non-trivial factors. A classical way to prove a system is prime is to show it has few self-joinings, that is, few T×invariant measures on X×X that project to μ. We show that there exists a prime transformation that has many self-joinings which are also large. In particular, its ergodic self-joinings are dense in its self-joinings and it has a self-joining that is not a distal extension of itself. As a consequence we construct the first known rank 1 transformation that is not quasi-distal and show that being quasi-distal is a meager property in the set of measure preserving transformations, which answers a question of Danilenko. This talk will not assume previous familiarity with joinings or prime transformations. This is joint work with Bryna Kra.

Seminar 3.7.19 Shahidi

Title: Area preserving diffeomorphisms with polynomial decay of correlations

Speaker: Farruh Shahidi (Pennsylvania State University)

Abstract: We show that any surface admits an area preserving C1+α diffeomorphism with non-zero Lyapunov exponents which is Bernoulli and has polynomial decay of correlations. We establish both upper and lower polynomial bounds on correlations (joint work with Ya. Pesin and S.Senti).

Seminar 2.28.19 Dymek

Title: Proximality of generalized -free systems

Speaker: Aurelia Dymek (Nicolaus Copernicus University)

Abstract: For any subset of integers by -free numbers we call the set of all integers that are not divisible by any element of . A -free system is the orbit closure of the characteristic function of -free numbers under the left shift. The study of -free systems began when Sarnak proposed to deal with dynamical properties of square-free system, i.e., -free system where is the set of all squares of primes. As he postulated this system is proximal. In the joint paper with Kasjan, Kulaga-Przymus and Lemanczyk we showed that a -free system is proximal if and only if contains an infinite pairwise coprime subset. Some multidimensional generalizations of -free systems where studied by Cellarosi, Vinogradov, Baake and Huck.

The topic of my talk is the proximality of generalized -free systems in the case of number fields and lattices. Our main results are the similar characterization of proximality in case of number fields and some lattices. We will give an example that such theorem fails in case of general lattices.

Seminar 2.21.19 Richter

Title: Uniform distribution of polynomial and non-polynomial sequences in nilmanifolds

SpeakerFlorian Richter (Northwestern University)

Abstract: The notion of uniform distribution conceptualizes the idea of a sequence of points that disperses evenly and proportionately throughout all parts of a mathematical space. The topic of my talk is the uniform distribution of a variety of polynomial and non-polynomial sequences in nilmanifolds, which are differentiable manifolds that possess a transitive nilpotent Lie group of diffeomorphisms. Our main results in this direction generalize the work of Leibman on the uniform distribution of polynomial orbits in nilmanifolds and the work of Frantzikinakis on the uniform distribution of nil-orbits along functions from a Hardy field. This also connects to open questions in arithmetic combinatorics and, in particular, to generalizations of Szemeredi’s theorem.

Seminar 1.31.19 Chen

TitleCr closing lemma for geodesic flows on Finsler surfaces

SpeakerDong Chen (Ohio State University)

Abstract: A Finsler metric on a smooth manifold is a smooth family of quadratically convex norms on each tangent space. The geodesic flow on a Finsler manifold is a 2-homogeneous Lagrangian flow. In this talk, I will give a proof of the Cr(r≥2) closing lemma for geodesic flows on Finsler surfaces.

The Cr closing lemma says that for any compact smooth Finsler surface and any vector v in the unit tangent bundle, the Finsler metric can be perturbed in Cr topology so that v is tangent to a periodic geodesic in the resulting metric. This allows us to get the density of periodic geodesics in the tangent bundle of a Cr generic Finsler surface.

Seminar 1.24.19 Son

Title: Uniform distribution of generalized polynomials and applications

Speaker: Younghwan Son (POSTECH, South Korea)

Abstract: Generalized polynomials are real-valued functions which are obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part. They form a natural extension of conventional polynomials, and appear, under different names, in a variety of mathematical contexts, from dynamics on nilmanifolds to number theory and mathematical games. Unlike the conventional polynomials, generalized polynomials may have quite intricate distributional properties. In this talk we will present recent results on uniform distribution of a large class of generalized polynomials and discuss some ergodic-theoretical applications. This is a joint work with Vitaly Bergelson and Inger Håland Knutson.

Seminar 1.17.19 Shi

TitleCr Closing lemma for partially hyperbolic diffeomorphisms on 3-manifolds

Speaker: Yi Shi (Peking University)

Abstract: The Cr-closing lemma is one well-known problem in the theory of dynamical systems. The problem is to perturb the original dynamical system so as to obtain a Cr-close system that has a periodic orbit passing through a given point. And this point is called Cr-closable. Steve Smale listed the Cr-closing lemma as one of mathematical problems for this century.

In this talk, we prove the Cr(r=2,3,⋯,∞) closing lemma for partially hyperbolic diffeomorphisms on 3-manifolds: every non-wandering point of these diffeomorphisms is Cr-closable. Moreover, we will show that Cr-generic conservative partially hyperbolic diffeomorphisms on 3-manifolds have dense periodic points.

Seminar 12.6.18 Donoso

Title: Quantitative multiple recurrence in ergodic theory

SpeakerSebastian Donoso (Universidad de O’Higgins)

Abstract: In this talk I will survey recent development of the multiple recurrence problem in ergodic theory. For a probability space (X,,μ) and measure preserving transformations T1,,Td, the problem is to study the largeness of the set of n such that


where a1,,ad take integer values on the integers and F is a suitable function. I will mention key results and comment on the problem for commuting transformations, linear and polynomial functions ai. I plan to provide some proofs that rely on combinatorial constructions.

Seminar 1.8.19 Wang

Title: Smooth Local Rigidity of Algebraic Actions

Speaker: Zhenqi Wang (Michigan State University)

Abstract: At first, we will introduce the background of algebraic actions and give some interesting examples. Next, we will review of various smooth rigidity results for higher-rank algebraic actions and recent progress. Finally, we will talk about the new progress on smooth rigidity of rank-one partially hyperbolic actions and future directions.

Seminar 11.1.18 Le

Title: Subsequences of multiple correlations

Speaker: Anh Le (Northwestern University)

Abstract: The results of Bergelson-Host-Kra and of Leibman say that a multiple correlation sequence can be decomposed as a sum of a nilsequence and a null sequence. Inspired by these results, Frantzikinakis asks the following question: Let rnrn be the sequence of primes, or [nc], or 2n. For a multiple correlation a(n), is it true that there exists a nilsequence b(n) and null sequence e(n) such that a(rn)=b(rn)+e(n)?

In this talk, I’ll briefly discuss why the answer is affirmative for the sequences of primes and [nc]. However, our main focus will be about 2n. The answer for this sequence is also yes and surprisingly related to the notion of sets of Bohr recurrence.