Exact dimension of dynamical stationary measures

TL;DR

This paper proves that the distributions of unstable and stable flag spaces for a certain class of random walks on SL_d(R) are exactly dimensional, providing precise geometric measure properties of these invariant measures.

Contribution

It establishes the exact dimensionality of stable and unstable flag space distributions for random walks on SL_d(R), a novel geometric measure result.

Findings

Distributions on flag spaces are exact dimensional.

Results apply to random walks with finite first moment and entropy.

Provides new insights into the geometric structure of invariant measures.

Abstract

We consider a random walk on $SL_d(\mathbb{R})$ with finite first moment and finite entropy. We show that the distributions of the unstable flag space and of the stable flag space are exact dimensional.