2025/11/01 by Ariel Rapaport, Haojie Ren, Rapaport, Ariel +1
Mathematics · #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2511.00729
Let θ be a finitely supported probability measure on SL(2,ℂ), and suppose that the semigroup generated by G:=supp(θ) is strongly irreducible and proximal. Let μ denote the Furstenberg measure on \mathbbCP1 associated to θ. Assume further that no generalized circle is fixed by all Möbius transformations corresponding to elements of G, and that G satisfies a mild Diophantine condition. Under these assumptions, we prove that dimμ=min\ 2,hRW/(2χ)\ , where hRW 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 \mathbbCP1 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.