Abstracts

Mike Hochman

Title: Directional minimality in algebraic systems.
Abstract: I will discuss a new result on the classification of invariant subshifts of irreducible zero-dimensional algebraic \(\mathbb{Z}^2\)-systems. In this setting the actions are never completely irreducible, and there always exist non-trivial subshifts: For instance, Einsiedler proved that there exist subsystems whose directional (topological) entropies take any value up to the maximal one. We show that if \(Y\) is a subshift of such a system, and the one-parameter subgroups of the shift action act minimally on \(Y\) (“\(Y\) is completely minimal”), then \(Y\) is finite. The talk will consist of a brief introduction and an outline of the proof, which is surprisingly simple.

Background: It will help if the audience is familiar with the definition of Ledrappier’s system, the definition of minimality for the \(\mathbb{Z}^2\) shift action, the notion of an expansive direction for a \(\mathbb{Z}^2\) action, and the statement of Boyle and Lind’s theorem that any infinite \(\mathbb{Z}^2\) system has a non-expansive direction [Expansive Subdynamics, Mike Boyle and Douglas Lind, Transactions of the American Mathematical Society Vol. 349, No. 1 (Jan., 1997), pp. 55-102]. I will state these in my talk, but quickly.

Slides (PDF)

Marlies Gerber

Title: Anti-Classification Results for the Kakutani Equivalence Relation.
Abstract: We prove that the Kakutani equivalence relation of ergodic invertible measure-preserving transformations is not a Borel set. This shows in a precise way that classification up to Kakutani equivalence is impossible. We also obtain this anti-classification result for ergodic area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action, as well as ergodic area-preserving real-analytic diffeomorphisms on the 2-torus. This work is joint with Philipp Kunde.

Slides (PDF)

Rodolfo Gutiérrez-Romo

Title: Coding the Teichmüller flow by veering triangulations.
Abstract: A half-translation surface is a collection of polygons whose sides are identified in pairs by translations and half-turns in such a way that the resulting topological surface is orientable. We consider two such collections to define the same half-translation surface if it is possible to cut the former along straight lines and reglue the pieces using the identifications to obtain the latter.

The Teichmüller flow is a geodesic flow on the space of half-translation surfaces. It can be defined directly on a polygon by stretching the horizontal direction and shrinking the vertical direction by the same factor. By a seminal result—independently by Masur and Veech in the early 1980s—the Teichmüller flow is known to be ergodic for the (unique) Lebesgue-class measure (nowadays it is even known to be exponentially mixing).

Since the work of Rauzy and Veech, a great deal of effort has been directed at finding suitable combinatorial schemes (or “codings”) to study the Teichmüller flow. In this talk, I will present a new coding leveraging the theory of veering triangulations. This is joint work with Mark Bell, Vincent Delecroix, Vaibhav Gadre and Saul Schleimer..

Slides (PDF)

Lewis Bowen

Title: Sofic entropy of subshifts.
Abstract: A group is sofic if it admits a sequence of partial actions on finite sets that approximates the action of the group on itself. This class of groups includes amenable groups and residually finite groups (such as free groups). Sofic entropy theory is a generalization of classical entropy theory to actions of sofic groups. This talk will be a quick introduction with an emphasis on symbolic dynamics. Near the end, I’ll discuss a new construction (joint with Dylan Airey and Frank Lin) of a mixing subshift of finite type over a virtually free group which has two different positive sofic entropies, depending on the choice of sofic approximation.

Slides (PDF)

Uri Gabor

Title: On the failure of Ornstein theory on the finitary category.
Abstract: The Ornstein’s isomorphism theorem (1970) asserts that i.i.d. processes with the same entropy are isomorphic. This was strengthened in 1979 by Keane and Smorodinsky who replaced the notion of isomorphism with the more strict notion of finitary isomorphism. Despite their success in lifting the Isomorphism theorem to the finitary category, the question of whether other classification theorems have finitary counterpart remained open. We introduce a new quantity assigned to a process, which is invariant under finitary isomorphism, and use it to show for three classification theorems the invalidity of their finitary counterpart: the preservation of being a Bernoulli shift through factors, Sinai’s factor theorem, and the weak Pinsker property. This gives negative answers to an old conjecture and a recent open problem..

Slides (PDF)

Anthony Quas

Title: Lyapunov exponents for Perron-Frobenius cocycles.
Abstract: I will introduce some joint work with Cecilia González-Tokman looking at stability and instability of Lyapunov exponents for Perron-Frobenius cocycles.

Slides (PDF)