Title: Tiling billiards and interval exchange transformations

Speaker: Diana 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.

Title: What’s new in rigidity

Speaker: Andrey 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.

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

Speaker: Victoria Sadovskaya (Pennsylvania State University)

Abstract: We consider a hyperbolic dynamical system (X,f) and a Holder continuous cocycle A over (X,f) with values in GL(d,ℝ), 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,ℝ)-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.

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

Speaker: Jon 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×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.

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).

Title: Proximality of generalized ℬ-free systems

Speaker: Aurelia Dymek (Nicolaus Copernicus University)

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

The topic of my talk is the proximality of generalized ℬ-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.

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

Speaker: Florian 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.

Title: Cr closing lemma for geodesic flows on Finsler surfaces

Speaker: Dong 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.

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.

Title: Cr 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.

Here is the current seminar schedule for Spring 2019:

Jan 17: Yi Shi (Peking University, China)

Jan 24: Younghwan Son (POSTECH, South Korea)

Jan 31: Dong Chen (Ohio State)

Feb 21: Florian Richter (Northwestern)

Feb 28: Aurelia Dymek (Nicolaus Copernicus University, Poland)

Mar 7: Farruh Shahidi (Penn State)

Mar 28: DOUBLE BILL:

2.00pm Jon Chaika (Utah)

3.00pm Victoria Sadovskaya (Penn State)

April 4: Andrey Gogolev (Ohio State)

April 18: Diana Davis (Swarthmore)