Probability and ergodic theory (PET) Ical Atom

For an ergodic measure preserving dynamical system $(X, \cal B, \mu, T)$ and a measurable function $f$ with values in $\mathbb{Z}$, we consider for $x \in X$ the set of values of the ergodic sums $S_nf(x):= \sum_0^{n-1} f(T^k x), n \geq 1$.

If $f$ is integrable with $\mu(f) > 0$, several properties of this set (from the point of view of recurrence or arithmetic sets) are simple consequences of Bourgain’s results (1989).

For example, the set ${S_nf(x), n \geq 1}$ contains infinitely many squares for a.e. $x$. If $f$ is not integrable, this property may fail, as shown by a construction of M. Boshernitzan. We give also a counter-example of an integrable centered function $f$ for which the cocycle $(S_nf(x), n \geq 1)$ is non regular and the property fails.

Time reversal symmetry arises in many dynamical systems. In particular, it is an important aspect of dynamical systems which emerge from physical theories such as classical mechanics, thermodynamics and quantum mechanics. In this talk, we introduce the notion of a reversible dynamical system in symbolic dynamics. We investigate conjugacy invariants of a topological Markov chain which possesses an involutory reversing symmetry.

The geometry of graphs is usually studied in the locally finite setup: each vertex has finitely many neighbors. By compactness arguments, one proves some useful and classical regularity theorems for such graphs. Such theorems are easily disproved for locally infinite graphs, but finding homogeneous counter-examples (transitive or Cayley) leads to interesting constructions. I will explain why the geometry of locally infinite graphs is worth studying, present my results, and state some questions I currently cannot answer.

In this talk I’ll present a joint work with Ariel Yadin, in which we solve the Furstenberg Entropy Realization Problem for finitely supported random walks (finite range jumps) on free groups and lamplighter groups. This generalizes a previous result of Bowen. The proof consists of several reductions which have geometric and probabilistic flavors of independent interests.

All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.

Given a Z^d shift of finite type and a nearest-neighbour interaction, we present sufficient conditions for efficient approximation of pressure and, in particular, topological entropy. Among these conditions, we introduce a combinatorial analog of the measure-theoretic property of Gibbs measures known as strong spatial mixing and we show that it implies many desirable properties in the context of symbolic dynamics. Next, we apply our results to some classical 2-dimensional statistical mechanics models such as the (ferromagnetic) Potts, (multi-type) Widom-Rowlinson, and hard-core lattice gas models for certain subsets of both the subcritical and supercritical regimes. The approximation techniques make use of a special representation theorem for pressure that may be of independent interest.

Part of this talk is joint work with Stefan Adams, Brian Marcus, and Ronnie Pavlov.

Chung and Li [Invent. Math. 2015] proved that for every expansive action of a countable polycyclic-by-finite group $\Gamma$ on a compact group $X$ by continuous group automorphisms, positive entropy implies the existence of non-diagonal asymptotic pairs. In the same paper they asked if the this holds in general for an expansive action of a countable amenable group $\Gamma$ on a compact space $X$.

In my talk I plan to explain the notions involved Chung and Li’s question and discuss a property of dynamical systems called the ``pseudo-orbit tracing property’’. R. Bowen introduced the pseudo-orbit tracing property in the 1970’s for $\mathbb{Z}$-actions while studying Axiom A maps. I will prove that Chung and Li’s question has an affirmative answer if one also assumes pseudo-orbit tracing, and explain implications for algebraic actions (automorphisms of compact abelian groups).

I will also explain why the answer to Chung and Li’s question is negative if one doesn’t assume the pseudo-orbit tracing property, even when the acting group is $\mathbb{Z}$, or when the action is algebraic (but not both).

Irrational rotations of the circle $T:\mathbb{R}/\mathbb{Z}\to \mathbb{R}/\mathbb{Z}$ are amongst the most studied dynamical systems. Rotations by badly approximable angels are exactly those for which the orbit of zero do not visit certain shrinking neighborhoods of zero, namely, there exists c>0 such that $T^n(0)\notin B\left(0,\frac{c}{n}\right)$ for all n. Khinchine proved that every orbit of any rotation of the circle misses a shrinking neighborhood of some point of the circle. In fact, he proved that the constant of these shrinking neighborhoods may be taken uniformly. The largest constant, however, remains unknown.

We will introduce the notion of approximation by rational functions in the field $\mathbb{F}_q((t-1)) ,$ formulate the analogue of Khinchine’s theorem over function fields and calculate the largest constant in this context.