Exact dimension of Furstenberg measures

TL;DR

This paper establishes the exact dimensionality of Furstenberg measures on flag spaces for certain discrete probability measures on SL d(R), under specific integrability and non-degeneracy conditions.

Contribution

It provides a precise dimension formula for Furstenberg measures in the discrete case, extending previous results to a broader class of measures.

Findings

Furstenberg measures are exact-dimensional under given conditions

Dimension formula derived for discrete measures on SL d(R)

Results apply under integrability and non-degeneracy assumptions

Abstract

For a probability measure $\mu$ on SL d (R), we consider the Furstenberg stationary measure on the space of flags. Under general non-degeneracy conditions, if $\mu$ is discrete and if g log g d$\mu$(g) < +$\infty$, then the measure $\nu$ is exact-dimensional.