# Dimensions of equilibrium measures on a class of planar self-affine sets

## TL;DR

This paper proves that equilibrium measures on certain planar self-affine sets are exact dimensional and their dimensions follow the Ledrappier-Young formula, linking entropy, Lyapunov exponents, and projections.

## Contribution

It establishes the exact dimensionality and dimension formula for K"aenm"aki measures on self-affine sets generated by diagonal and anti-diagonal matrices with strong separation.

## Key findings

- Measures are exact dimensional.
- Dimension satisfies Ledrappier-Young formula.
- K"aenm"aki measure decomposes into Gibbs measures.

## Abstract

We study equilibrium measures (K\"aenm\"aki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the K\"aenm\"aki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.