TITLES AND ABSTRACTS

Sarah Bray
Ergodic geometry for nonstrictly convex Hilbert geometries

Abstract:

The geodesic flow of a compact hyperbolic Riemannian manifold is a classical object of study, dating back to the independent work of Hedlund and Hopf in the 30s. Since then, geodesic flows arising from manifolds generalizing notions of compact, Riemannian, and hyperbolic have entertained dynamicists to this day.

Strictly convex Hilbert geometries nicely generalize hyperbolic Riemannian geometries and the geodesic flow on quotients has been well-studied by Benoist, Crampon, Marquis, and others. In contrast, nonstrictly convex Hilbert geometries in 3 dimensions have a fascinating geometric irregularity which forces a very different approach for studying the dynamics of the geodesic flow. In this talk, we present techniques in the intersection of Hilbert geometries, nonuniform hyperbolicity theory, and ergodic geometry, culminating in the construction of a measure of maximal entropy which is ergodic for the geodesic flow of a compact 3-dimensional Hilbert geometry.

Jerome Buzzi
Smooth surface diffeomorphisms and their invariant probability measures

Abstract:

A classical result of Katok shows that surface diffeomorphisms are approximated by horseshoes with respect to topological entropy.  From a recent result of Hochman, one obtains Borel conjugacy to a Markov shift respecting all invariant ergodic probability measures except possibly for measures with zero entropy and measures maximizing the entropy at their periods. I will explain what joint works with Boyle and with Crovisier and Sarig say (and don't say) about these latter measures and present some open problems.

Nishant Chandgotia
An Introduction to Hom-shifts

Abstract:

The study of multidimensional shift of finite type is mired by a host of undecidable questions. However things become a little better if we assume a little symmetry and isotropy: Let H be a finite undirected graph. By Z^d we mean the Cayley graph of Z^d with respect to the standard generators. Denote by Hom(Z^d, H), the space of adjacency preserving maps from Z^d  to H; it can be naturally thought of as a nearest neighbour shift of finite type. In this talk, we will explore many aspects of these shift spaces, hinting at (time permitting) how undecidability comes into the picture.

Maria Isabel Cortez

Topological orbit equivalence, dimension groups and eigenvalues

 Van Cyr

Automorphisms of zero entropy dynamical systems

Abel Farkas

How to extend results about self-similar sets to attractors of graphdirected systems and subshifts of finite type with no effort

Matthew Foreman
Global Structure Theorems

Abstract:

This lecture addresses two central problems in classical ergodic theory: the classification problem and the realization problem. An historical focus of ergodic theory has been the structure and properties of single transformations. Perhaps the most prominent is the Furstenberg-Zimmer structure theorem which describes ergodic transformations in terms of limits of compact and weakly mixing extensions.

This lecture discusses a new phenomenon, Global Structure Theorems. We define two categories: the Odometer Based Systems (finite entropy transformations that have an odometer factor) and Circular Systems (those diffeomorphisms built using a version of the Anosov-Katok technique.) The morphisms  in each category are measure-theoretic joinings.

The main result is that these two categories are isomorphic by a composition-preserving bijection that that takes conjugacies to conjugacies, extensions to extensions, weakly mixing extensions to weakly mixing extensions, compact extensions to compact extensions, distal towers to distal towers (and more). In short, all of the structure present in the odometer based systems is exactly reflected in the Circular Systems and vice versa.

The lecture will conclude with a provocative conjecture.

This is joint work with B. Weiss.

Thomas French

Follower and Extender Sets in One-Dimensional Shifts

Abstract:

In a one-dimensional shift space, the follower set of a word w is  the set of all right-infinite sequences which may legally follow w. The follower set sequence records the number of distinct follower sets corresponding to words of length n in X. Extender sets are a generalization of follower sets and the extender set sequence is defined similarly. This talk will explore connections and contrasts between the follower and extender set sequences and the complexity sequence of a shift space X.


Su Gao
Rank-one transformations: topological conjugacy and isomorphism

Abstract:

I will talk about rank-one transformations as both topological dynamical systems and measure-preserving transformations. I will give a complete characterization of the topological conjugacy in terms of the defining parameters. For the isomorphism problem of rank-one transformations as measure-preserving transformations, I will focus on canonically bounded systems with commensurate parameters. We will characterize when such transformations are isomorphic, disjoint, or neither. This is joint work with Aaron Hill.

Andrey Gogolev

New partially hyperbolic diffeomorphisms

Abstract:

Due  to close ties to stable ergodicity and robust transitivity, partial hyperbolicity plays an important role in smooth dynamics. It is interesting to know which manifolds and which homotopy classes of diffeomorphisms admit partially hyperbolic representatives. Recently two new methods were developed for constructing new partially hyperbolic diffeomorphisms:
-- Anosov (and partially hyperbolic) bundles;
-- Twisting of Anosov flows.
We will present an overview of these methods and explain how they lead to abundance of mapping classes with partially hyperbolic representatives. We will also explain how these two methods can be combined to work together.

This lecture represents work of many people including Tom Farrell, Pedro Ontaneda, Federico Rodriguez Hertz, Christian Bonatti, Kamleh Parwani, Rafael Potrie, Andy Hammerlindl.

Boris Kalinin
Normal forms on non-uniformly contracting foliations.

Abstracft:

We consider a diffeomorphism f of a compact manifold M which contracts an invariant foliation W  with smooth leaves. If the differential of f  on TW  has narrow band spectrum, there exist coordinates H _x: W_x --> T_x W  in which f_W(x) is a polynomial in a finite-dimensional Lie group G. We construct H_x that depend smoothly on x along the leaves of W and give an atlas with transition maps in G. Our results apply, in particular, to C¹-small perturbations of algebraic systems. Further, we construct polynomial normal forms for smooth extensions of measure preserving systems by non-uniform contractions and obtain the above results for the stable "foliation" of an arbitrary measure preserving diffeomorphism f. This yields an f-invariant structure of a G homogeneous space on almost every leaf.


Alexander S Kechris
Descriptive Graph Combinatorics

Abstract:

This talk is about a relatively new subject, developed in the last two decades or so, which is at the interface of descriptive set theory and graph theory but also has interesting connections with other areas such as ergodic theory and probability theory.

The object of study is the theory of definable graphs, usually Borel or analytic, on Polish spaces and one investigates how combinatorial concepts, such as colorings and matchings, behave under definability constraints, i.e., when they are required to be definable or perhaps well-behaved in the topological or measure theoretic sense.

Zhiqiang Li

Ergodic theory for expanding Thurston maps 

Ryo Moore

Double recurrence Wiener-Wintner theorem and weighted nonconventional ergodic averages

Abstract:

We will first discuss the extension of J. Bourgain's double recurrence theorem to the Wiener-Wintner averages, and a further extension to the polynomial case. Secondly, we will discuss that this extension of the double recurrence theorem can be used to obtain a new good universal weight for nonconventional ergodic averages. This is a joint work with I. Assani, and partly with D. Duncan.

Anima Nagar
Strongly Transitive Systems

Abstract:

One of the oldest concepts in the study of dynamical systems has been `transitivity'. Its stronger forms -- minimality, mixing, weak mixing -- have also been well studied. In this talk, we look into yet another stronger form which we call Strongly transitive'. These are systems (X,f) for which the backward orbit, {y: f^n(y) = x, for some  positive integer n},  is dense in X, for all x in X. We study  properties of such systems and discuss some questions arising from this study that are still unanswered.

This talk is based on joint works with Akin, Auslander and Kannan.

Volodia Nekrashevych
Palindromic subshifts and simple groups of intermediate growth

Abstract:

We will show how any non-free minimal action of the infinite dihedral group on the Cantor set can be used to construct infinite finitely generated periodic groups. Actions of low complexity (linearly repetitive) produce groups of intermediate growth. In particular, one can construct simple groups of intermediate growth, existence of which was a long standing open question.

Ronnie Pavlov
One-sided almost specification and intrinsic ergodicity for subshifts

Abstract:

The specification property has been fundamental in the study of topological  dynamical systems since its introduction by Bowen, and in particular is known to imply intrinsic ergodicity, i.e. uniqueness of the measure of maximal entropy.

For subshifts, almost specification is a weakening of specification which allows for concatenation of arbitrary words in the language if a small  number of letters are changed in each, parametrized by a mistake function  g(n) = o(n). It was recently shown that even almost specification for the constant function g(n) = 4 does not imply intrinsic ergodicity.

In joint work with Vaughn Climenhaga, we've defined a variant of this property called one-sided almost specification. I'll discuss some of our results, including the fact that one-sided almost specification for any constant mistake function g(n) does imply intrinsic ergodicity. 

Jesse Peterson

T.B.A.

Ian Putnam
New minimal homeomorphisms, why they shouldn't exist and why it's interesting that they do

Abstract:

An old question is: which infinite, compact metric spaces admit minimal (uniquely ergodic) homeomorphisms? Probably the most famous positive answer is the result of Fathi and Hermann that all spheres of odd dimension (at least 3) admit minimal, uniquely ergodic diffeomorphisms. On the other hand, perhaps the most remarkable negative result is the Hopf-Lefschetz Theorem that asserts for continuous self-maps of 'nice'  spaces, a non-vanishing Lefschetz number implies the existence of a fixed-point. Here, we present  an infinite, compact metric space whose cohomology is the same as that of a point. So any continuous self-map will have Lefschetz number 1. We also show that this space admits a minimal, uniquely ergodic homeomorphism. This has important consequences in George Elliott's classification program for C*-algebras. We will discuss these in general terms, assuming no familiarity with C*-algebras.  This is joint work with Robin Deeley (Hawaii) and Karen Strung (IMPAN, Poland).

Ilkka Torma

Structure and computability of multidimensional shift spaces

 
Ilya Vinogradov
Effective equidistribution of horocycle lifts

Abstract:

We give a rate in the problem of equidistribution of lifted horocycles in the space of unimodular two-dimensional lattice translates.  The ineffective version is due to Elkies and McMullen and relies on Ratner's Theorem.  The approach used here builds on recent works of Strombergsson and of Browning and the author.

 
Agnieszka Zelerowicz

Thermodynamics of some non-uniformly hyperbolic attractors