Modul: MAT760 Ergodic theory and dynamical systems seminar

## A dynamical proof of the Shmerkin—Wu theorem

Talk by Prof. Dr. Tim Austin

Speaker invited by: Prof. Dr. Artur Avila

**Date:** 11.12.23 **Time:** 15.00 - 16.00 **Room:** Y27H25

Let a < b be multiplicatively independent integers, both at least 2. Let A,B be closed subsets of [0,1] that are forward invariant under multiplication by a, b respectively, and let C be A x B. An old conjecture of Furstenberg asserted that any line not parallel to either axis must intersect C in Hausdorff dimension at most max(dim C,1)−1. He was able to prove a partial result in this direction using a new class of measure-valued processes, now referred to as "CP chains". A few years ago, Shmerkin and Wu independently gave two different proofs of Furstenberg's conjecture. In this talk I will sketch a more recent third proof that builds on some of Furstenberg's original results. In addition to those, the main ingredients are a version of the Shannon—McMillan—Breiman theorem relative to a factor and some standard calculations with entropy and Hausdorff dimension.''