Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure

TL;DR

This paper proves the exact dimensionality of the Furstenberg measure and establishes a Ledrappier-Young formula for its dimension under certain algebraic conditions, advancing the understanding of measure dimensions in linear group actions.

Contribution

It extends the theory of measure dimensions by proving exact dimensionality and deriving a Ledrappier-Young formula for Furstenberg measures under strong irreducibility and proximality.

Findings

Furstenberg measure is exact dimensional under specified conditions.

A Ledrappier-Young type formula for the measure's dimension is established.

The proof adapts Feng's approach for self-affine measures.

Abstract

Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also establish a Ledrappier-Young type formula for its dimension. The general strategy of the proof is based on the argument given by Feng for the exact dimensionality of self-affine measures.