# Equilibrium States for Expanding Thurston Maps

Equilibrium States for Expanding Thurston Maps In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $${\mu_\phi}$$ μ ϕ for each expanding Thurston map $${f : S^2\rightarrow S^2}$$ f : S 2 → S 2 together with a real-valued Hölder continuous potential $${\phi}$$ ϕ . Here the sphere S 2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials that are co-homologous up to a constant, and that the measure-preserving transformation f of the probability space $${(S^2,\mu_\phi)}$$ ( S 2 , μ ϕ ) 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, we recover various results in the literature for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

# Equilibrium States for Expanding Thurston Maps

, Volume 357 (2) – Jan 31, 2018
62 pages

/lp/springer_journal/equilibrium-states-for-expanding-thurston-maps-fCGuGrTtr2
Publisher
Springer Journals
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-3073-9
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $${\mu_\phi}$$ μ ϕ for each expanding Thurston map $${f : S^2\rightarrow S^2}$$ f : S 2 → S 2 together with a real-valued Hölder continuous potential $${\phi}$$ ϕ . Here the sphere S 2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials that are co-homologous up to a constant, and that the measure-preserving transformation f of the probability space $${(S^2,\mu_\phi)}$$ ( S 2 , μ ϕ ) 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, we recover various results in the literature for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.

### Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Jan 31, 2018

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