Welcome to the online working seminar in ergodic theory
A bit about the seminar:
It meets weekly online on Tuesdays at 2 pm Utah time.
Participants take turns presenting a paper, which hopefully admits a nice
one hour presentation.
This seminar is flexible, so if one needs an hour
and a half, or wishes to give a pre-talk to make it more accessible
that is great.
Title: Boshernitzan's criterion for unique ergodicity.
Speaker: Jon Chaika
Additional info: There will be a pre-talk from 1:20-1:50 Mountain time
to introduce symbolic dynamics and linear block growth.
Title: Caroline Series' The modular surface and continued fractions
Speaker: Claire Merriman
Additional info: Case of the Gauss map
Title: An introduction to naive entropy
There are simple formulas defining "naive entropy" for continuous/measure preserving actions of a countable group G
on a compact metric/probability space.
It turns out that if G is amenable, then this naive entropy
coincides with topological/Kolmogoro-Sinai entropy of the action,
while for non-amenable groups both naive entropies take only two values:
0 or infinity. During my talk,
I will try to sketch the proofs of these facts.
I will follow: T. Downarowicz, B. Frej, P.-P. Romagnoli,
Shearer's inequality and infimum rule for Shannon entropy and
Dynamics and numbers, 63-75, Contemp. Math., 669,
Amer. Math. Soc., Providence, RI, 2016. MR3546663 and
P. Burton, Naive entropy of dynamical systems.
Israel J. Math. 219 (2017), no. 2, 637-659. MR3649602.
Additional information: Dominik will be giving a pre-talk at 1:20 pm Utah
time on Amenable groups, their actions and entropy.
Title: Rigidity of Geodesic Planes in Hyperbolic Manifolds
Speaker: Osama Khalil
Abstract: We present a proof of the following fact: the image of a totally geodesic immersion of the hyperbolic plane into a compact hyperbolic 3-manifold is either closed or dense. This result is due to Shah and Ratner independently. We will discuss the proof due to Shah which builds on the work of Margulis in the proof of the Oppenheim conjecture. The talk will serve as an introduction to the technique of polynomial divergence of unipotent flows which is fundamental in homogeneous dynamics. We do not assume prior knowledge of the subject.
Link to Shah's paper
Title: Veech's Criterion for a process to be prime
Slides. Corrections from the slides I used in the talk are in red.
Abstract: This talk will present Veech's criterion for an ergodic
probability measure preserving system to be prime. It will define
factors of measure preserving systems, prime and self-joinings and provide
examples. It uses disintegration of measures,
the ergodic decomposition and Haar's Theorem. It will state these
results and have examples of disintegration of measures and
the ergodic decomposition, but wont discuss their proofs.
Title: Bratteli-Vershik models for Cantor and Borel dynamical systems
Abstract: In this talk we will introduce Bratteli diagrams and Vershik maps.
Herman-Putnam-Skau proved that for every minimal Cantor dynamical system
there exists a Bratteli-Vershik model.
We will discuss the proof of this theorem,
some of its applications and recent developments.
We will also discuss Bratteli-Vershik models for Borel dynamical systems
Finally, we will briefly talk about connections between
Bratteli diagrams and
flows on translation surfaces (Lindsey-Treviño).
Title: Unique ergodicity of horocycle flows on compact quotients of SL(2,R)
Speaker: Jon Chaika
Video Note the video starts a minute or so into the talk.
Abstract: Furstenberg proved that the horocycle flow on any compact quotient of
SL(2,R) is uniquely ergodic.
This has been generalized by many people. I will present a proof due to
Yves Coudène, which I find elegant and can prove some
of the generalizations of Furstenberg's theorem too.
Title:Return-time sets for probability preserving transformations
Speaker: Nishant Chandgotia
Abstract: Abstract: Given a probability preserving system (X, \mu, T) and a set U of positive measure contained in X we denote by N(U,U) the set of integers n such that the measure of
U intersected with T^n(U) is positive. These sets are called return-time sets and are of very special nature. For instance, Poincaré recurrence theorem tells us that the set must have bounded gaps while Sarkozy-Furstenberg theorem tells us that it must have a square. The subject of this talk is a very old question (going back to Følner-1954 if not earlier) whether they give rise to the same family of the sets as when we restrict ourselves to compact group rotations. This was answered negatively by Kříž in 1987 and recently it was proved by Griesmer that a return-time set need not contain any translate of a return-time set arising from compact group rotations. In this talk, I will try to sketch some of these proofs and give a flavour of results and questions in this direction.
Title: The counting formula of Eskin and McMullen
Speaker: Paul Apisa
Abstact:Given a lattice acting on the hyperbolic plane, how many orbits of a point intersect the ball the radius of r as r gets big? Similarly, given a hyperbolic surface with a geodesic gamma, how many lifts of gamma to the hyperbolic plane intersect the ball of radius r? Using the mixing of geodesic flow on hyperbolic surfaces, Eskin and McMullen found a short beautiful argument to find the asymptotics for these counting questions (and more general ones on affine symmetric spaces). The key insight is to relate the counting problems to the equidistribution of circles under geodesic flow. In this talk I will discuss how to deduce circle equidistribution and counting problem asymptotics from mixing. The talk will involve many pictures and focus on the case of hyperbolic surfaces, however, the arguments presented will be general and their application to counting on general affine symmetric spaces will be explained at the end of the talk.
Eskin and McMullen's paper
Title:Specification and the measure of maximal entropy
Speaker: Vaughn Climenhaga
Abstract: There are various proofs that a transitive uniformly hyperbolic dynamical system has a unique measure of maximal entropy. I will outline a proof due to Bowen that uses the specification and expansivity properties, focusing on the example of shift spaces. If time permits, I will describe how Bowen's proof works for equilibrium states associated to nonzero potential functions.
Link to the "board notes" Vaughn will produce.