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

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.