On the Ledrappier-Young formula for self-affine measures

TL;DR

This paper establishes conditions under which self-affine measures in the plane satisfy the Ledrappier-Young formula, linking entropy, Lyapunov exponents, and dimension, with applications to specific classes of measures.

Contribution

It provides new sufficient conditions for self-affine measures to meet the Ledrappier-Young formula, extending previous results and offering alternative proofs for known theorems.

Findings

Self-affine measures satisfy the Ledrappier-Young formula under strong separation and dominated splitting conditions.

Dimensions of such measures equal the Lyapunov dimension under certain criteria.

Application to measures generated by lower triangular matrices and alternative proof of Hueter-Lalley's theorem.

Abstract

Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier-Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some applications, namely, we give another proof for Hueter-Lalley's theorem and we consider self-affine measures and sets generated by lower triangular matrices.