Title: Solution to the inverse of Sarnak’s conjecture

Speaker: Tomasz Downarowicz (Wroclaw University, Poland)

Department of Mathematics

Title: Solution to the inverse of Sarnak’s conjecture

Speaker: Tomasz Downarowicz (Wroclaw University, Poland)

Title: Weakened specification properties and intrinsic ergodicity for subshifts

Speaker: Ronnie Pavlov (University of Denver)

Abstract: The specification property has been fundamental in the study of topological dynamical systems since its introduction by Bowen, and among other properties, implies intrinsic ergodicity, i.e. uniqueness of the measure of maximal entropy. For subshifts, specification amounts to the ability to concatenate arbitrary pairs of n-letter words given a uniform gap (i.e. independent of n).

In this setting, several natural weakenings of specification can be defined. For instance, one can allow non-uniform gaps, or allow a small number of “edits” to the words rather than including a gap. In both cases, one can parametrize by an auxiliary function f(n) controlling gap size/number of edits.

I will summarize some recent results in this area, beginning with examples showing that several weakened specification properties do not imply intrinsic ergodicity even for slowly growing f(n), and finishing with recent joint work with Vaughn Climenhaga, in which we defined a new weakened specification property which does imply intrinsic ergodicity for small enough f(n).

Title: Two types of KAM-nondegenerate nearly integrable systems with positive metric entropy

Speaker: Dong Chen (Penn State)

Abstract: The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still have a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws lots of attention. In this talk I will present two types of C^\infty small Lagrangian perturbation of the geodesic flow on a flat torus. Both resulting flows have positive metric entropy, from which we get positive metric entropy outside some KAM tori. What is special in the second type is that positive metric entropy comes from an arbitrarily small tubular neighborhood of one trajectory. This is a joint work with D. Burago and S. Ivanov.

Title: On pointwise multiple convergence problems

Speaker: Wenbo Sun (The Ohio State University)

Abstract: Multiple convergence problems is a topic that has been widely studied in ergodic theory. These questions usually have interesting applications in combinatorics. While the L2L2 multiple convergence problems is well-understood by now, little is known for the pointwise convergence problems. In this talk, I will discuss about some recent advances in the pointwise convergence problems. This is joint work with Sebastian Donoso.

Title: Diophantine approximation problems for groups of toral automorphisms

Speaker: Vladimir Finkelshtein (University of Illinois at Chicago)

Abstract: I will present sharp rates for a shrinking target problem for the action of an arbitrary subgroup of SL(2,Z) on the 2-torus. This can also be viewed as a noncommutative Diophantine approximation problem. The methods require construction of spectrally optimal random walks on groups acting properly cocompactly on Gromov hyperbolic spaces. Additionally, I will explain how similar estimates for this problem in higher dimension can be obtained using harmonic analysis.

Speaker: Keith Burns (Northwestern)

Title: Mixing properties of the Weil-Petersson geodesic flow

Abstract: I will talk about the geodesic flow for the Weil-Petersson metric on the moduli space of a surface that supports hyperbolic metrics. This is a Riemannian metric with negative sectional curvatures. However the classical results of Anosov do not apply because the metric is incomplete and the sectional curvatures and their derivatives are not uniformly bounded. It was not until the 21st century that this geodesic flow was shown to be mixing (and in fact Bernoulli).

I will give some ideas from the proof and also from more recent work directed towards showing that the flow is exponentially mixing in case of the torus with one puncture.

This is joint work with Howie Masur, Carlos Matheus, and Amie Wilkinson

Here is our provisional schedule of talks for September and October 2016. More talks will be confirmed soon for the rest of semester. All talks are in MW154 at 3.00pm-4.00pm.

Sep 1: Dan Thompson (Ohio State)

Sep 15: Keith Burns (Northwestern)

Sep 22: Vladimir Finkelshtein (UIC)

Oct 6: Wenbo Sun (Ohio State)

Oct 20: Dong Chen (Penn State)

Nov 3 COLLOQUIUM TALK: Laura DeMarco (Northwestern)

DOUBLE BILL MINI-CONFERENCE NOV 10:

1.50pm -2.50pm: Tomasz Downarowicz (Wroclaw University, Poland)

3.00pm-3.50pm: Ronnie Pavlov (Denver)

Speaker: Dan Thompson (Ohio State)

Title: Generalized beta-transformations and the entropy of unimodal maps

Abstract: Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation x↦ beta x (mod1), where beta > 1, and replacing some of the branches with branches of constant negative slope. We would like to describe the set of beta for which these maps can admit a Markov partition. We know that beta (which is the exponential of the entropy of the map) must be an algebraic number. Our main result is that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. This extends an analysis of Solomyak for the case of beta-transformations, who obtained a sharp bound of the golden mean in that setting.

I will also describe a connection with some of the results of Thurston’s fascinating final paper, where the set of all conjugates of numbers arising as exponential of the entropy for some post-critically finite unimodal map is shown to describe an intriguing fractal. These numbers are included in the setting that we analyze.