Title: Unique Equilibrium States for Geodesic Flows on Surfaces without Focal Points

Speaker: Nyima Kao (University of Chicago)

Abstract: It is well-known that for compact uniformly hyperbolic systems Hölder potentials have unique equilibrium states. However, it is much less known for non-uniformly hyperbolic systems. In his seminal work, Knieper proved the uniqueness of the measure of maximal entropy for the geodesic flow on compact rank 1 non-positively curved manifolds. A recent breakthrough made by Burns, Climenhaga, Fisher, and Thompson which extended Knieper’s result and showed the uniqueness of the equilibrium states for a large class of non-zero potentials. This class includes scalar multiples of the geometric potential and Hölder potentials without carrying full pressure on the singular set. In this talk, I will discuss a further generalization of these uniqueness results, following the scheme of Burns-Climenhaga-Fisher-Thompson, to equilibrium states for the same class of potentials over geodesic flows on compact rank 1 surfaces without focal points. This work is an MRC project joint with Dong Chen, Kiho Park, Matthew Smith, and Régis Varão.

Title: The Erdos sumset conjecture

Speaker: Florian Richter (Ohio State University)

Abstract: A longstanding open conjecture of Erdos states that every subset of the integers with positive density contains a sum B+C of two infinite sets B and C. I will talk about recent joint work with Joel Moreira and Donald Robertson in which we resolve this conjecture. Our poof utilizes ideas and methods coming from Ergodic Theory, including Bergelson’s intersectivity lemma, various decomposition theorems of arithmetic functions into structured and pseudo-random components, and some borrowed techniques from Beiglboeck’s proof of Jin’s theorem.

Title: A classification of intermediate β-transformations

Speaker: Tony Samuel (Cal Poly San Luis Obispo)

Abstract: In this talk we consider transformations of the unit interval of the form $\beta x + \alpha \bmod{1}$ where $1 < \beta < 2$ and $0 \leq α \leq 2 – \beta$.  These transformations are called intermediate β-transformations.  We will discuss some old and new results concerning these transformations, for instance, their kneading sequences, their absolutely continuous invariant measures and dynamical properties such as topological transitivity and the sub-shift of finite type property.  Moreover, we address how the kneading sequences and absolutely continuous invariant measures change as we let $(\beta, \alpha)$ converge to $(1, \theta)$, for some $\theta \in [0, 1]$. Finally, some open problems and applications of these results to one-dimensional Lorenz maps and quasicrystals will be alluded to.

Title: Limiting behavior of multiple polynomial averages

Speaker: Andreas Koutsogiannis (Ohio State University)

Abstract: The study of the norm limiting behavior of multiple ergodic averages has been of great importance in the area of ergodic theory. A central result, Szemerédi’s theorem (i.e., every subset of natural numbers of positive upper density contains arbitrary long arithmetic progressions) follows by a classical result on multiple ergodic averages due to Furstenberg. In this talk we will mainly deal with averages along integer part of special families of real polynomials, for a single transformation (recent joint work with D. Karageorgos) as well as for multiple commuting transformations; we will refer to results along other integer valued sequences mainly due to Bergelson, Knutson, Leibman, Chu, Frantzikinakis, Host and Kra. We will also sketch how to obtain by the aforementioned results, the corresponding results along prime numbers.

Title: Periodic point growth for C^2 maps of the two sphere

Speaker: Yun Yang (CUNY)

Abstract: There are two basic mechanisms producing periodic orbits in a dynamical systems: contraction (via the Contraction mapping theorem, Banach fixed point theorem) and degree (via topological methods such as the Lefschetz theorem).These two mechanisms play an important role in the joint work of Enrique Pujals, Michael Shub and myself with assistance from Sylvain Crovisier on Shub’s conjecture: Let f : M -> M be a C2 map of a compact manifold. Then the exponential growth rate in fixed points of fn bounded below by the growth rate of the Lefschetz numbers of f^n. In this talk, I will present the proof of this conjecture in the case where f: S2 -> S2 has positive entropy and reverses orientation in the direction of vanishing exponents.

Title: On the relationship between entropy and periodic orbits

Speaker: Bryce Weaver (Xavier University)

Abstract: Via Pesin theory, complexity, measured by entropy, is inexorably linked to structure, namely stable/unstable manifolds. Another such link is the relationship between entropy and growth of periodic orbits. One pioneer result in this area was by G. Margulis in the late ‘60s for geodesics flows on negatively curved manifolds. We explore an adaptation of the technique established by G. Margulis in the case of case of geodesic flows on manifolds with regions of positive curvature. We then close with some settings where there is potential for similar adaptations, in particular the Bunimovich stadium billiards.

Title: Lee-Yang zeros for the Cayley Tree and expanding maps of the circle

Speaker: Roland Roeder (IUPUI)

Abstract: I will explain how to use detailed properties of expanding maps of the circle (Shub-Sullivan rigidity, Ledrappier-Young formula, large deviations principle,…) to study the limiting distribution of Lee-Yang zeros for the Ising Model on the Cayley Tree. No background in mathematical physics is expected of the audience. This is joint work with Ivan Chio, Anthony Ji, and Caleb He.

Title: The automorphism group of a zero entropy symbolic system

Speaker: Van Cyr (Bucknell)

Abstract: The symmetries of a symbolic dynamical system X form an interesting and often complicated group called its automorphism group. Although this group is always countable, it is frequently extremely complex for positive entropy subshifts (containing free subgroups, the fundamental group of every 2-manifold, and every finite group). By contrast, the group of automorphisms of a zero entropy subshift is often considerably more tame and it has been possible to prove a number of strong algebraic results. In this talk I will discuss some of these results and open problems.

Title: Recurrence along some sequences involving primes

Speaker: Younghwan Son (Pohang University of Science and Technology, Korea)

Abstract: One of the important themes in ergodic theory is the phenomena of recurrence, which concerns how the initial state returns to the original state. The ergodic method introduced by Furstenberg has been used to deduce the combinatorial structures inherent in large subsets of integers by proving recurrence statements in dynamical systems.

In this talk, we will present some general results on uniform distribution involving primes to establish new recurrence statements, which refine the previous results obtained by Sarkozy and Furstenberg. This is a joint work with V. Bergelson and G. Kolesnik.

Title: Badly approximable points on manifolds and unipotent orbits in homogeneous spaces

Speaker: Lei Yang (Sichuan University, China)

Abstract: We will study n-dimensional badly approximable points on manifolds. Given an smooth non-degenerate submanifold in R^n, we will show that any countable intersection of the sets of weighted badly approximable points on the manifold has full Hausdorff dimension. This strengthens a previous result of Beresnevich by removing the condition on weights and weakening the analytic condition on manifolds to smooth condition. Compared with the work of Beresnevich, we study the problem through homogeneous dynamics. It turns out that the problem is closely related to the study of distribution of long pieces of unipotent orbits in homogeneous spaces.

Title: Generic properties of extensions

Speaker: Michael Schnurr (Max-Planck-Institut, Germany)

Abstract: Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not have an intermediate nil-factor.

Title: Marked length rigidity for Fuchsian buildings

Speaker: Dave Constantine (Wesleyan University)

Abstract: Suppose we are given two metrics on a space and told that for every element of the fundamental group of the space, the length of the shortest curve representing it for each metric is the same. Must the two metrics be the same? This is the marked length spectrum (MLS) rigidity problem. Most famously, the answer is `yes’ for negatively-curved Riemannian metrics on closed surfaces, yet the problem remains wide open for negatively curved metrics in higher dimensions.

In this talk I’ll discuss joint work with Jean-Francois Lafont proving some MLS rigidity results for Fuchsian buildings. The proof uses the combinatorial structure of the buildings, as well as an extension of the technology developed to prove MLS rigidity for surfaces which we must carefully adapt to overcome the non-surface-like behavior of the buildings.

Here is the schedule of talks for Spring semester 2018. We have also scheduled some talks in “May-mester”.  All talks are on Thursdays in MW154 at 3.00pm-4.00pm (unless otherwise indicated).

Spring 2018:

January 18: Dave Constantine (Wesleyan) (joint with GGT seminar)

January 25: Michael Schnurr (Max-Planck-Institut, Germany)

February 1: Lei Yang (Sichuan University, China)

February 8: Younghwan Son (Pohang University, Korea).

February 22: Van Cyr (Bucknell)

March 1: Roland Roeder (IUPUI)

March 8: Bryce Weaver (Xavier University)

March 17: Aaron Brown (University of Chicago) [Invited Speaker, AMS sectional meeting at Ohio State]

March 22: Yun Yang (City University of New York)

April 5: Andreas Koutsogiannis (Ohio State)

April 19: Tony Samuel (Cal State Poly San Luis Obispo)

May-Mester 2018:

May 10: Florian Richter (Ohio State)

May 31: Nyima Kao (University of Chicago)