Abstract: I will discuss a quantitative equidistribution result for the random walk on a torus arising from the action of the group of affine transformations. This is a joint work with Weikun He and Elon Lindenstrauss.

Abstract: I will report on recent results on sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This is joint work with F. Pene. This result allows to also obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case.

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Abstract: We construct a set which is dense in the Bohr topology on the group of integers and which is not a set of measurable recurrence, answering a question asked by Bergelson, Hegyvári, Ruzsa, and the author, in various combinations.  This talk will provide a broad overview and explain details of the construction. We will see similarities to many other examples in additive combinatorics and ergodic theory, such as Igor Kriz's construction showing topological recurrence does not imply measurable recurrence, and Ruzsa's niveau sets.

Abstract: Quantitative equidistribution for linear random walks on the torus was first obtained by Bourgain, Furman, Lindenstrauss and Mozes. In this talk I will present a recent progress where the proximality assumption in their result is relaxed. I will also discuss an application to expansion in groups. This is based on a joint work with Nicolas de Saxcé.

Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=3c855e5a-44bf...

Abstract: I'll discuss  various  "ratio mixing"  properties
of transformations preserving infinite measures e.g. "Krickeberg mixing" (based on the example in Hopf's 1936 book) & "rational weak mixing". I'll also introduce a new one connected to "tied down" renewal theory.

Contains joint works with Hitoshi Nakada, Dalia Terhesiu & Toru Sera


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Abstract: One of the first things we learn about a (proper) Gromov hyperbolic geodesic space X is the construction of the visual boundary of X. An ergodic theorist then learns that for a non-elementary discrete group of isometries G acting properly on X, there is an interesting family of \delta_G-quasi-conformal measures on the boundary. The parameter \delta_G is called the critical exponent of G, and is equal to the exponential growth rate of the orbit Gx in X.

Abstract: A circle packing is a canonical way of representing a planar graph. There is a deep connection between the geometry of the circle packing and the proababilistic property of recurrence/transience of the simple random walk on the underlying graph, as shown in the famous He-Schramm Theorem. The removal of one of the Theorem's assumptions - that of bounded degrees - can cause the theorem to fail. However, by using certain natural weights that arise from the circle packing for a weighted random walk, (at least) one of the directions of the He-Schramm Theorem remains true.

Title: Equidistribution of the image in the torus of sparse points on dilating analytic curves

Abstract: Let G be a countable group. A mutliorder is a collection of 
bijections from G to Z (the integers) on which G acts by a special 
"double shift". If G is amenable, we also require some uniform Folner 
property of the order intervals. The main thing is that mutiorder exists 
on every countable amenable group, which can be proved using tilings.
For now, multiorder provides an alternative formula for entropy of a 
process and we are sure in the nearest future it will allow at produce 

E-Seminar.

Abstract: A circle packing is a canonical way of representing a planar graph. There is a deep connection between the geometry of the circle packing and the proababilistic property of recurrence/transience of the simple random walk on the underlying graph, as shown in the famous He-Schramm Theorem. The removal of one of the Theorem's assumptions - that of bounded degrees - can cause the theorem to fail. However, by using certain natural weights that arise from the circle packing for a weighted random walk, (at least) one of the directions of the He-Schramm Theorem remains true.

Abstract: Host proved a strengthening of Rudolph and Johnson's measure rigidity theorem: if a probability measure is invariant, ergodic and has positive entropy for the map x2 mod 1, then a.e. point equidisitrbutes under x3 mod 1. Host also proved a version for toral endomorphisms, but its hypotheses are in some ways too strong, e.g. it requires one of the maps to be expanding, so it does not apply to pairs of  automorphisms. In this talk I will present an extension of Host's result (almost) to its natural generality.

Abstract:
A restricted permutation of a locally finite directed graph $G=(V,E)$ is a vertex permutation $\pi: V\to V$ for which $(v,\pi(v))\in E$, for any vertex $v\in V$. The set of such permutations, denoted by $\Omega(G)$, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. In the particular case presented by Schmidt and Strasser (2016), where $V=\mathbb{Z}^d$ and $(n,m)\in E$ iff $(n-m)\in A$ ($A\subseteq \mathbb{Z^d}$ is fixed), $\Omega(G)$ is a subshift of finite type.