Dimension of Furstenberg measures on $\mathbb{CP}^{1}$

TL;DR

This paper determines the dimension of Furstenberg measures on complex projective space under certain conditions, linking it to random walk entropy and Lyapunov exponents, and overcoming challenges posed by the space's real dimension.

Contribution

It establishes a formula for the dimension of Furstenberg measures on P^1 in terms of random walk entropy and Lyapunov exponent, using new methods for the real 2-dimensional setting.

Findings

Dimension formula: P^1 measure dimension equals min{2, h_{RW}/(2\u03c7)}.

Proved under mild Diophantine and irreducibility conditions.

Analysis addresses challenges of real dimension 2 in a projective, contracting setting.

Abstract

Let $\theta$ be a finitely supported probability measure on $\mathrm{SL}(2,\mathbb{C})$, and suppose that the semigroup generated by $\mathcal{G}:=\mathrm{supp}(\theta)$ is strongly irreducible and proximal. Let $\mu$ denote the Furstenberg measure on $\mathbb{CP}^{1}$ associated to $\theta$. Assume further that no generalized circle is fixed by all M\"obius transformations corresponding to elements of $\mathcal{G}$, and that $\mathcal{G}$ satisfies a mild Diophantine condition. Under these assumptions, we prove that $\dim\mu=\min\left\{ 2,h_{\mathrm{RW}}/\left(2\chi\right)\right\} $, where $h_{\mathrm{RW}}$ and $\chi$ denote the random walk entropy and Lyapunov exponent associated to $\theta$, respectively. Since our result expresses $\dim\mu$ in terms of the random walk entropy rather than the Furstenberg entropy, and relies only on a mild Diophantine condition as a separation assumption, we are forced to directly confront difficulties arising from the ambient space $\mathbb{CP}^{1}$ having real dimension $2$ rather than $1$. Moreover, our analysis takes place in a projective, contracting-on-average setting. This combination of features introduces significant challenges and requires genuinely new ideas.