Equilibrium measures of self-affine fractals

Abstract

Classical fractals such as the Sierpinski gasket, middle-third Cantor set and Menger sponge have the property of being self-similar: each of these sets is equal to a finite union of rescaled and repositioned copies of itself.

Moreover, given any finite collection of contracting similitudes there exists a unique nonempty compact set which is similar to itself via that set of contractions. Under suitable separation conditions (which guarantee that the rescaled copies of the fractal do not overlap too much) the dimension theory of the self-similar fractals defined by such collections of contractions has been well-understood since the 1980s.

When the rescaling maps are allowed to be non-conformal, linearly distorting contractions rather than similarity transformations the situation changes drastically, and while a unique compact nonempty limit set always exists, few dimension results are known. In this talk I will describe some of my recent results in this area including joint work with Pablo Shmerkin, Antti Kaenmaki and Jairo Bochi.