Equilibrium states of generalised singular value potentials and applications to affine iterated function systems

TL;DR

This paper characterizes equilibrium states for a class of potentials related to affine iterated function systems, showing their properties and implications for the dimension theory of self-affine fractals.

Contribution

It provides a complete description of equilibrium states for these potentials, including bounds on their number and their support properties, and applies these results to problems in fractal dimension theory.

Findings

Number of ergodic equilibrium states is bounded by a dimension-dependent constant.

All equilibrium states are fully supported and satisfy a Gibbs inequality.

Removing a map from the IFS strictly reduces the affinity dimension.

Abstract

We completely describe the equilibrium states of a class of potentials over the full shift which includes Falconer's singular value function for affine iterated function systems with invertible affinities. We show that the number of distinct ergodic equilibrium states of such a potential is bounded by a number depending only on the dimension, answering a question of A. K\"aenm\"aki. We prove that all such equilibrium states are fully supported and satisfy a Gibbs inequality with respect to a suitable subadditive potential. We apply these results to demonstrate that the affinity dimension of an iterated function system with invertible affinities is always strictly reduced when any one of the maps is removed, resolving a folklore open problem in the dimension theory of self-affine fractals. We deduce a natural criterion under which the Hausdorff dimension of the attractor has the same strict reduction property.