A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

TL;DR

This paper establishes a partial converse to Hutchinson's theorem, showing that self-affine measures with maximal Hausdorff dimension imply the system is essentially composed of similarity transformations, under certain conditions.

Contribution

It proves that affine iterated function systems supporting self-affine measures with maximal Hausdorff dimension must be composed of similarity transformations, unless they are reducible.

Findings

Equilibrium measures of affine IFS are not Bernoulli unless reducible or similarity-based.

Supports a partial converse to Hutchinson's theorem.

Uses thermodynamic formalism and spectral properties of linear groups.

Abstract

A well-known theorem of J.E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier results in the thermodynamic formalism of affine iterated function systems due to Bochi, Feng, K\"aenm\"aki, Shmerkin and the first named author and also relies on the work of Benoist on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.