Frostman dimension of Furstenberg measure for $\mathrm{SL}(2,\mathbb{R})$ random matrix products

TL;DR

This paper derives a formula for the Frostman dimension of Furstenberg measures associated with certain random matrix products in SL(2,R), and analyzes spectral gaps of transfer operators near zero.

Contribution

It provides a new explicit formula for the Frostman dimension and characterizes spectral gaps of transfer operators for these matrix products.

Findings

Frostman dimension formula for Furstenberg measure

Spectral gap existence near zero for transfer operators

Conditions of strong irreducibility and proximality

Abstract

For compactly supported $\mu \in \mathcal{P}(\mathrm{SL}(2,\mathbb{R}))$ satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on H\"older spaces in this setting.