Dynamical Systems

Last week, Balázs Bárány, István Kolossváry, and I organised a focussed workshop at the Erdős Center (Rényi Institute) in Budapest. The event was centred on recent progress on self-conformal and self-similar sets, in particular analytic iterated function systems and the Fourier decay of invariant measures.

The purpose of this blog entry is two-fold:

1. To tell a little more about the event.
2. To collect the notes, slides, and open problems that came out of it.

A few days ago, I uploaded a joint article with Balázs Bárány and Henna Koivusalo to arXiv. The article is called Dynamical covering sets in self-similar sets and I will briefly summarise the paper here.

TLDR: Imagine you are trying to cover the circle $\mathbb{S}^1$ with randomly centred balls $B(x_k,r_k)$, where $x_k$ is distributed uniformly on $\mathbb{S}^1$ and $r_k$ is a decreasing sequence. The Dvoretzky covering question¹ asks for conditions on $r_k$ for the whole circle to be covered almost surely. Similarly, one can ask for when the cover is of full measure, or what size the covering set is.

In this article, we study an analogous dynamical problem for symbolic balls (cylinders) in self-similar sets, where we pick our initial point according to Bernoulli measures. We obtain a single pressure formula that describes the entire dimension theory, which shows distinct regions to the (known) homogeneous case.