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

Abstract

Let $\hat{I}¸$ be a finitely supported probability measure on $\mathrm{SL}(2,\mathbb{C})$, and suppose that the semigroup generated by $\mathcal{G}:=\mathrm{supp}(\hat{I}¸)$ is strongly irreducible and proximal. Let $\hat{I}¼$ denote the Furstenberg measure on ${\mathbb{CP}}^1$ associated to $\hat{I}¸$. Assume further that no generalized circle is fixed by all Möbius transformations corresponding to elements of $\mathcal{G}$, and that $\mathcal{G}$ satisfies a mild Diophantine condition. Under these assumptions, we prove that $\dim \hat{I}¼=\min \left\{2,h_{\mathrm{RW}}/\left(2\ddot{I}‡\right)\right\}$, where $h_{\mathrm{RW}}$ and $\ddot{I}‡$ denote the random walk entropy and Lyapunov exponent associated to $\hat{I}¸$, respectively. Since our result expresses $\dim \hat{I}¼$ 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.

55 pages. Minor changes in the introduction; all other parts unchanged