Seminar 12.3.20 Frantzikinakis

Title: Furstenberg Systems of Bounded Sequences

Speaker: Nikos Frantzikinakis – University of Crete

Abstract: Furstenberg systems are measure preserving systems that are used to model statistical properties of bounded sequences of complex numbers. They offer a different viewpoint for a variety of problems for which progress can be made by a partial or complete description of suitably chosen Furstenberg systems. In this lecture I will give several examples of this principle and in the process we will see several structural properties of Furstenberg systems arising from smooth functions and bounded multiplicative functions.

Seminar 11.19.20 Hafuta

Title: Limit theorems for time dependent expanding dynamical systems

Speaker: Yeor Hafuta – Ohio State University

Abstract: 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 as 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 Rulle-Perron-Frobenius theorem, which yields that the variance grows linearly fast when the underlying maps are close enough to a single expanding map.

Pdf of slides available here

Title: On the dimension drop conjecture for diagonal flows on the space of lattices

Speaker: Shahriar Mirzadeh – Michigan State University

Abstract: (see attached pdf for better formatting): Consider the set of points in a homogeneous space X=G/Γ whose gt-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when has real rank 1.

In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: G=SLm+n(R), Γ=SLm+n(Z)and gt=\diag(et/m,⋯,et/m,e−t/n,⋯,e−t/n) We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.

Zoom recording available here

Pdf of slides available here

Seminar 10.29.20 Kucherenko

Title: Multiple phase transitions on compact symbolic systems

Speaker: Tamara Kucherenko – City College of New York

Abstract: A first-order phase transition refers to a loss of differentiability of the pressure function with respect to a parameter regarded as the inverse temperature. Such non-differentiability necessarily implies coexistence of several equilibrium states, although the converse is not true. In the case of H\”older continuous potentials on transitive SFTs the pressure is real analytic and there are no phase transitions. Therefore, in order to allow the possibility of phase transitions one needs to consider potentials that are merely continuous. Note that the convexity of the pressure implies that a continuous potential has at most countably many phase transitions. We show that the case of infinitely many phase transitions can indeed be realized. In this talk we present a method to explicitly construct a continuous potential on a full shift with an infinite number of first order phase transitions occurring at any increasing sequence of predetermined points. This is based on joint work with Anthony Quas and Christian Wolf.

Zoom recording available here

Seminar 10.22.20 Best

Title: The Furstenberg-Sárközy theorem and asymptotic total ergodicity

Speaker: Andrew Best – Ohio State University

Abstract: The Furstenberg-Sárközy theorem asserts that the difference set E-E of a subset E of the natural numbers with positive upper density contains a (nonzero) square. Furstenberg’s approach relies on a correspondence principle and a version of the Poincaré recurrence theorem along squares; the latter is shown via the result that for any measure-preserving system $(X,\mathcal{B},\mu,T)$ and set A with positive measure, the ergodic average $\frac{1}{N} \sum_{n=1}^N \mu(A \cap T^{-n^2}A)$ has a positive limit c(A) as N tends to infinity. Motivated — by what? we shall see — to optimize the value of c(A), we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings (Z/N_m Z) is asymptotically totally ergodic if and only if the least prime factor of N_m grows to infinity. From this fact, we derive some combinatorial consequences. These results are based on joint work with Vitaly Bergelson.

Zoom recording available here

Seminar 10.16.20 Wang

Title: Central Limit Theorem for equilibrium measures in dynamical systems

Speaker: Tianyu Wang – Ohio State University

Abstract: Central limit theorem of certain class of equilibrium measures is a heavily studied statistical property in smooth dynamics. In the first half of the talk, I will briefly introduce some common strategies to study CLT that are useful in many classic settings, e.g. Anosov flows, expanding maps on the unit circle, (countable) Markov shift, etc. In the second half, I will show how specification can be applied to derive an asymptotic version of CLT for the equilibrium measures in the case of geodesic flow on non-positively curved rank-one manifold. This method is first introduced by Denker, Senti, Zhang and the result is based on a recent joint work with Dan Thompson.

Zoom recording available here

Seminar 10.9.20 Demers

Title: Thermodynamic Formalism for Sinai Billiards

Speaker: Mark Demers – Fairfield University

Abstract: While the ergodic properties of Sinai billiards with respect to the SRB measure are well understood, there have been few studies of other invariant measures and equilibrium states. As a step in this direction, we study the family of geometric potentials $– t \log (J^uT)$, $t>0$. For any finite horizon Sinai billiard map $T$, we find $t_* >1$ such that for each $t \in (0, t_*)$, there exists a unique equilibrium state $\mu_t$ for the potential. We show that $\mu_t$ is exponentially mixing for H\”older observables, and that the pressure function $P(t)$ is analytic on $(0,t_*)$. This extends our recent results for the case $t=0$, corresponding to the measure of maximal entropy. This is joint work with Viviane Baladi.

Zoom recording available here

Pdf of slides available here

Seminar 10.2.20 Das

Title: Successive minima of lattice trajectories and topological games to compute fractal dimensions

Speaker: Tushar Das – University of Wisconsin – La Crosse

Abstract: We present certain sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establishes a new connection to the metric theory via a variational principle that computes the fractal dimensions of various sets of number-theoretic interest. Applications of our results include computing the Hausdorff and packing dimensions of the set of singular systems of linear forms and showing they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of the correspondence principle (initiated by Dani, and deepened by Kleinbock and Margulis), the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Highlights of our program include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry. I hope to present a sampling of open questions and directions that have yet to be explored, some of which may be pursued by either following or adapting the technology described in my talk.

Zoom recording available here

Pdf of slides available here

Seminar 9.18.20 Kao

Title: Pressure Metrics for Deformation Spaces of Quasifuchsian Groups with Parabolics

Speaker: Lien-Yung “Nyima” Kao – George Washington University

Abstract: Thurston pointed out that one can use variations of lengths of closed geodesics on hyperbolic surfaces to construct a Riemannian metric on the Teichmueller space. When the surface is closed, Wolpert showed that Thurston’s construction recovers the Weil-Petersson metric. Using thermodynamic formalism, McMullen proposed a new perspective to this Riemannian metric, and called it the pressure metric. In this talk, I will discuss how to extend this dynamical construction to spaces of quasiconformal deformations of (non-compact) finite area hyperbolic surfaces. This is a joint work with Harry Bray and Dick Canary.

Zoom recording available here

Pdf of slides available here

New Ohio State Online Ergodic Theory Seminar

UPDATE: We will continue our program in Spring 2021. However, we are taking a brief Winter hiatus. We expect to resume in February.

We are pleased to announce that we will be running an online seminar program in Fall 2020. The seminar will take place in our usual time slot unless otherwise noted – Thursdays 3.00pm (EST). Some seminars are scheduled at an alternate time of Friday 12.40pm (EST).

Our current schedule for the semester follows:

Sept 17: Lien-Yung “Nyima” Kao (George Washington University)

Oct 2 (Friday, 1pm EST): Tushar Das (University of Wisconsin)

Oct 9 (Friday, 12.40pm EST): Mark Demers (Fairfield University)

Oct 16 (Friday, 12.40pm EST): Tianyu Wang (Ohio State)

Oct 22: Andrew Best (Ohio State)

Oct 29: Tamara Kucherenko (City College of New York)

Nov 12: Shahriah Mirzadeh (Michigan State)

Nov 19: Yeor Hafuta (Ohio State)

Dec 3: Nikos Frantzikinakis (University of Crete)