There will be an ergodic theory meeting at the University of Warwick on Wednesday 22 November 2023. This is part of an LMS Scheme 3 funded network involving the following universities: Birmingham, Bristol, Durham, Exeter, Loughborough, Manchester, Open, Queen Mary, St Andrews and Warwick. For information about other meetings in the series, see
https://people.maths.bris.ac.uk/~matmj/LMSergodicmeetings.html
If you are from a university in the network and would like to request financial support to attend, please contact Thomas Jordan at thomas.jordan@bristol.ac.uk
All talks will take place in room MS.03 of the Zeeman Building. (Click here for a downloadable campus map.) After entering the Zeeman Building, go up the stairs in front of you and then another flight of stairs to the second floor to reach MS.03 (there is also a lift near the Zeeman entrance). A group of people will go for lunch at 12:30pm, meeting by the Zeeman Building main entrance. There will be an early dinner after the talks.
Schedule:
2:00pm-3:00pm: Terry Soo (UCL)
Independent, but not identically distributed coin-flips
3:00pm-3:30pm: Tea break
3:30pm-4:30pm: Tim Austin (Warwick)
A dynamical proof of the Shmerkin—Wu theorem
4:30pm-4:45pm: Short break
4:45pm-5:45pm: Henna Koivusalo (Bristol)
Shrinking targets on self-affine sets
Abstracts:
Tim Austin: A dynamical proof of the Shmerkin—Wu theorem
Abstract: 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 $A \times B$. 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.
Henna Koivusalo: Shrinking targets on self-affine sets
Abstract: The classical shrinking target problem concerns the following set-up: Given a dynamical system $(T, X)$ and a sequence of targets $(B_n)$ of $X$, we investigate the size of the set of points $x$ of $X$ for which $T^n(x)$ hits the target $B_n$ for infinitely many $n$. In this talk I will discuss shrinking target problems in the context of iterated function systems, where `size' is studied from the perspective of dimension. I will give an overview of the topic, with the aim to, by the end, cover an upcoming result on geometric shrinking targets on Przytycki-Urbanski-type affine iterated function systems. Analysing this particular model requires heavy use of the theory of Bernoulli convolutions. This talk is based on a work joint with Thomas Jordan.
Terry Soo: Independent, but not identically distributed coin-flips
Abstract: In joint work with Zemer Kosloff, we will discuss the dynamical properties of a seemingly innocuous perturbation of a sequence of independent and identically distributed (iid) coin-flips to one that is no longer stationary. In the stationary case, Ornstein proved that iid systems are completely classified up to isomorphism by their Shannon entropy. We will find that in the nonstationary case, the usual entropy theory no longer applies, but we will recover an explicit version of the Sinai factor theorem that allows us to generate iid randomness from a nonstationary source.
