2014/10/18 by Zhiqiang Li, Li, Zhiqiang · 2 citations
Mathematics · #37B99 #37D20 #37D25 #37D35 (Primary) #37D40 #37D50 #37F15 #57M12 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1410.4920
openalex publication_date 2014/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state μϕ for each expanding Thurston map f: S2→ S2 together with a real-valued Hölder continuous potential ϕ. Here the sphere S2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials which are co-homologous upto a constant, and that the measure-preserving transformation f of the probability space (S2,μϕ) is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of f, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, all the above results hold for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.