# 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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9277306795120239, "perplexity": 949.147505253788}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510969.31/warc/CC-MAIN-20181017023458-20181017044958-00132.warc.gz"}