Dimension of Furstenberg measures on $\mathbb{CP}^{1}$
Ariel Rapaport, Haojie Ren
TL;DR
This work determines the dimension of the Furstenberg measure μ on CP^1 associated to a finitely supported θ on SL(2, C) under strong irreducibility, proximality, a mild Diophantine separation (weak Diophantine), and no fixed generalized circle. The authors express dim μ in terms of the random walk entropy h_RW and the Lyapunov exponent χ via dim μ = min{2, h_RW/(2χ)}, avoiding direct reliance on Furstenberg entropy and exploiting only a mild separation condition. The proof develops a projective, contracting-on-average framework and introduces a novel entropy-increase mechanism that propagates entropy under convolution from G to C via a linearization, together with a Ledrappier–Young-type formula for exact dimensionality. The result extends dimension theory for stationary fractal measures to the 2-dimensional CP^1 setting and broadens the scope beyond previously discrete or one-dimensional regimes, with potential implications for random matrix-product dynamics and fractal geometry in higher dimensions.
Abstract
Let $θ$ be a finitely supported probability measure on $\mathrm{SL}(2,\mathbb{C})$, and suppose that the semigroup generated by $\mathcal{G}:=\mathrm{supp}(θ)$ is strongly irreducible and proximal. Let $μ$ denote the Furstenberg measure on $\mathbb{CP}^{1}$ associated to $θ$. 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 $\dimμ=\min\left\{ 2,h_{\mathrm{RW}}/\left(2χ\right)\right\} $, where $h_{\mathrm{RW}}$ and $χ$ denote the random walk entropy and Lyapunov exponent associated to $θ$, respectively. Since our result expresses $\dimμ$ 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.