Pure Maths Colloquium: Ariel Rapaport

A compact \(K\subset\mathbb{R}^{2}\) is called self-affine if there exist affine contractions \(\varphi_{1},...,\varphi_{m}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\) such that \(K=\cup_{i=1}^{m}\varphi_{i}(K)\). In the 1980s, Kenneth Falconer introduced a value \(s\), called the affinity dimension, which is the ‘‘expected’’ value for the dimension of \(K\). I will discuss a recent project with Mike Hochman, building on our joint work with Balázs Bárány, in which we establish the equality \(\dim K=s\) under mild assumptions.