# Entropy and dimension of disintegrations of stationary measures

## TL;DR

This paper generalizes results on the dimensionality of stationary measures from SL_2(R) to GL(R^d), linking measure dimensions to Lyapunov exponents and entropy, and exploring their relation to spectral simplicity.

## Contribution

It extends the understanding of stationary measure disintegrations to higher dimensions, connecting entropy, Lyapunov exponents, and spectral properties.

## Key findings

- Dimensions of conditional measures relate to Lyapunov exponent gaps.
- Entropy measures are connected to Lyapunov spectrum simplicity.
- Results generalize previous SL_2(R) findings to GL(R^d).

## Abstract

We extend a result of Ledrappier, Hochman, and Solomyak on exact dimensionality of stationary measures for $\text{SL}_2(\mathbb{R})$ to disintegrations of stationary measures for $\text{GL}(\mathbb{R}^d)$ onto the one dimensional foliations of the space of flags obtained by forgetting a single subspace. The dimensions of these conditional measures are expressed in terms of the gap between consecutive Lyapunov exponents, and a certain entropy associated to the group action on the one dimensional foliation they are defined on. It is shown that the entropies thus defined are also related to simplicity of the Lyapunov spectrum for the given measure on $\text{GL}(\mathbb{R}^d)$.