Some observations on K\"aenm\"aki measures

TL;DR

This paper studies properties of K"aenm"aki measures, providing conditions for Lyapunov exponent gaps, bounds on ergodic measures in low dimensions, and posing an open problem on Hausdorff dimensions of self-affine measures.

Contribution

It offers a simple sufficient condition for Lyapunov exponent gaps, sharp bounds on ergodic measures in dimensions up to 4, and introduces an open problem on Hausdorff dimension related to matrix semigroups.

Findings

Provided a sufficient condition for gaps between Lyapunov exponents.

Established sharp bounds for the number of ergodic K"aenm"aki measures in dimensions up to 4.

Posed an open problem on the Hausdorff dimension of self-affine measures.

Abstract

In this note we investigate some properties of equilibrium states of affine iterated function systems, sometimes known as K\"aenm\"aki measures. We give a simple sufficient condition for K\"aenm\"aki measures to have a gap between certain specific pairs of Lyapunov exponents, partially answering a question of B. B\'ar\'any, A. K\"aenm\"aki and H. Koivusalo. We also give sharp bounds for the number of ergodic K\"aenm\"aki measures in dimensions up to 4, answering a question of J. Bochi and the author within this range of dimensions. Finally, we pose an open problem on the Hausdorff dimension of self-affine measures which may be reduced to a statement concerning semigroups of matrices in which a particular weighted product of absolute eigenvalues is constant.