Title: Local limit theorem for certain ballistic random walks in random
Abstract: We study the model of random walks in random environments in
dimension four and higher under Sznitman's ballisticity condition (T').
We prove a version of a local Central Limit Theorem for the model and also
the existence of an equivalent measure which is invariant with respect
to the point of view of the particle. This is a joint work with Noam Berger
and Moran Cohen.
Manchester building, Hebrew University of Jerusalem, (Room 209)
Title: Topological structures and the pointwise convergence of some averages for commuting transformations
Abstract: ``Topological structures'' associated to a topological dynamical
system are recently developed tools in topological dynamics. They have
several applications, including the characterization of topological
dynamical systems, computing automorphisms groups and even the pointwise
convergence of some averages. In this talk I will discuss some developments
of this subject, emphasizing applications to the pointwise convergence of
(joint work with Françoise Dal'Bo and Andrea Sambusetti) Given a finitely generated group G acting properly on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. In this work we are interested in the following question: what can we say if H and G have the same exponential growth rate? This problem has both a combinatorial and a geometric origin.
Abstract: A permutation representation of a group G is called highly transitive if it is transitive on k-tuples of points for every k. Until just a few years ago groups admitting such permutation representations were thought of as rare. I will focus on three rather recent papers: G-Garion, Hall-Osin, Gelander-G-Meiri (in preparation) showing that such groups are in fact very common.
For a given deterministic measure we construct a random measure on the Brownian path that has expectation the given measure. For the construction we introduce the concept of weak convergence of random measures in probability. The machinery can be extended to more general sets than Brownian path.
To a linear differential equation on the projective line with finitely many points of singularities, is associated a monodromy group; when the singularities are "reguar singular", then the monodromy group gives more or less complete information about the (asymptotics of the ) solutions.
The cases of interest are the hypergeometric differential equations, and there is much recent work in this area, centred around a question of Peter Sarnak on the arithmeticity/thin-ness of these monodromy groups. I give a survey of these recent results.