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

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 work in the thermodynamic formalism of affine iterated function systems due to Feng, Käenmäki, Bochi and the first named author and also relies on the work of Benoist and Quint on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

“…Suppose that the image of Γ is not-relatively compact in PGL 𝑑 (ℝ). As 𝜌(Γ) < GL 𝑑 (ℝ) is (strongly) irreducible, it follows that the semigroup 𝜌(Γ) is 𝑟-proximal for some 𝑟 ∈ {1, … , 𝑑 − 1} (this is standard, see, e.g., [56,Lemma 3.6]). It then follows by the same argument in Lemma 5.1 that (𝑀 𝑗 𝑛 ) is 𝑟-contracting for each 𝑗 = 1, … , 𝑚.…”

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

“…Since every such measure has support equal to the attractor of the IFS, this leads directly to the desired lower bound in the classical self-similar case of Hutchinson's article [33]. For irreducible affine IFS which do not consist of similitudes a self-affine measure with the desired Lyapunov dimension cannot exist, and indeed the supremum of the Lyapunov dimensions of the possible selfaffine measures is necessarily strictly smaller than the affinity dimension [40]. Theorem 1.1(i) may therefore not be applied directly to deduce (ii), but is applied indirectly via an additional result as follows.…”