Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension

Abstract

A fundamental problem in the dimension theory of self‐affine sets is the construction of high‐dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such high‐dimensional measures is to investigate measures of maximal Lyapunov dimension; these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by Falconer. While the existence of these equilibrium states has been well known for some years their structure has remained elusive, particularly in dimensions higher than two. In this article we give a complete description of the equilibrium states of the singular value function in the three‐dimensional case, showing in particular that all such equilibrium states must be fully supported. In higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states. As a corollary, giving a solution to a folklore open question in dimension three, we prove that for a typical self‐affine set in R 3 , removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension.