Table Of ContentAtlantis Studies in Dynamical Systems
Series Editors: H. Broer · B. Hasselblatt
Zhiqiang Li
Ergodic Theory
of Expanding
Thurston Maps
Atlantis Studies in Dynamical Systems
Volume 4
Series editors
Henk Broer, Groningen, The Netherlands
Boris Hasselblatt, Medford, USA
The“Atlantis StudiesinDynamicalSystems”publishes monographs inthearea of
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Zhiqiang Li
Ergodic Theory of Expanding
Thurston Maps
ZhiqiangLi
Institute for Mathematical Sciences
StonyBrookUniversity
StonyBrook, NY
USA
Atlantis Studies inDynamical Systems
ISBN978-94-6239-173-4 ISBN978-94-6239-174-1 (eBook)
DOI 10.2991/978-94-6239-174-1
LibraryofCongressControlNumber:2017930418
©AtlantisPressandtheauthor(s)2017
Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorbyany
means, electronic or mechanical, including photocopying, recording or any information storage and
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Printedonacid-freepaper
To the loving memory of my grandmother
Fengxian Shen
Preface
This monograph came out of my thesis work under the supervision of my Ph.D.
advisorMarioBonkduringmygraduatestudiesattheUniversityofMichigan,Ann
Arbor, and later at the University of California, Los Angeles. It focuses on the
dynamics,morespecificallyergodictheory,ofsomecontinuousbranchedcovering
maps on the 2-sphere, called expanding Thurston maps.
More than 15 years ago, Mario Bonk and Daniel Meyer became independently
interestedinsomebasicproblemsonquasisymmetricparametrizationof2-spheres,
related to the dynamics of rational maps. They joined forces during their time
together at the University of Michigan and started their investigation of a class of
continuous(butnotnecessarilyholomorphic)mapsmodelingasubclassofrational
maps. These maps belong to a bigger class of continuous maps on the 2-sphere
studied by William P. Thurston in his famous characterization theorem of rational
maps (see [DH93]). As a result, Mario Bonk and Daniel Meyer called their maps
expanding Thurston maps. Related studies were carried out by other researchers
around the same time, notably Peter Haïssinsky and Kevin Pilgrim [HP09], and
James W. Cannon, William J. Floyd, and Walter R. Parry [CFP07].
Bylate2010,MarioBonkandDanielMeyerhadsummarizedtheirfindingsina
reader-friendly arXiv draft [BM10] entitled ExpandingThurstonmaps,whichthey
initially intended to publish in the AMS Mathematical Surveys and Monographs
series. In order to make the material even more accessible, they decided later to
expand their draft. This led to a long delay for the final published version [BM17]
with almost twice the size of [BM10].
IwasintroducedtoexpandingThurstonmapsbyMarioBonksoonafterIjoined
in the graduate program at the University of Michigan. I quickly got deeply fas-
cinatedbythissubjectduetotheconnectionstogeometry,analysis,combinatorics,
and dynamical systems.
I finished my first project on the periodic points and properties of the measures
of maximal entropy of expanding Thurston maps under the supervision of Mario
Bonk(laterappearedin[Li13],seeChap.4)afterwemovedtoLosAngeles.Ithen
decidedtocontinueworkingontheergodictheoryofexpandingThurstonmaps,on
which I eventually wrote my thesis.
vii
viii Preface
Thismonographcoversinvestigationsonthemeasuresofmaximalentropy,and
moregenerally,equilibrium statesofexpandingThurstonmaps,andtheirrelations
to the periodic points and the preimage points. In order to study the equilibrium
states,thetheoryofthermodynamicalformalismforHöldercontinuouspotentialsis
established in our context (see Chap. 5). The study of equidistribution results also
leadstoacloseinvestigationontheexpansionpropertiesofourdynamicalsystems
(see Chap. 6) and the discovery of some large deviation results (see Chap. 7).
This monograph is also intended to serve as a basic reference for the theory
of thermodynamical formalism in our context. The applications to the study of the
dynamical zeta functions were also kept in mind when this monograph was being
prepared.Assuch,complex-valuedfunctionspaces areusedwheneverthey donot
introduce too much complication.
Acknowledgments I want to express my deep gratitude to Mario Bonk for intro-
ducing me to this subject, and for his encouragements, guidance, and strict stan-
dards. I would also like to thank Ilia Binder, Nhan-Phu Chung, Peter Haïssinksy,
DanielMeyer,andKevinPilgrimforhelpfulconversations.Mygratitudealsogoes
to our editors Boris Hasselblatt, Henk W. Broer, and Arjen Sevenster for their
patient support and helpful comments. I want to acknowledge the partial supports
from NSF grants DMS-1162471 and DMS-1344959. Last but not least, I want to
thank my wife, Xuan Zhang, for her understanding, support, and love.
Stony Brook, NY, USA and Hangzhou, China Zhiqiang Li
December 2016
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
2 Thurston Maps . .... .... ..... .... .... .... .... .... ..... .... 15
2.1 Historical Background ..... .... .... .... .... .... ..... .... 16
2.2 Definition for Thurston Maps.... .... .... .... .... ..... .... 17
2.3 Cell Decompositions.. ..... .... .... .... .... .... ..... .... 18
2.4 Notions of Expansion for Thurston Maps... .... .... ..... .... 22
2.5 Visual Metric ... .... ..... .... .... .... .... .... ..... .... 23
2.6 Invariant Curves. .... ..... .... .... .... .... .... ..... .... 25
3 Ergodic Theory. .... .... ..... .... .... .... .... .... ..... .... 29
3.1 Covers and Partitions . ..... .... .... .... .... .... ..... .... 29
3.2 Entropy and Pressure . ..... .... .... .... .... .... ..... .... 30
3.3 The Ruelle Operator for Expanding Thurston Maps... ..... .... 32
3.4 Weak Expansion Properties . .... .... .... .... .... ..... .... 34
4 The Measure of Maximal Entropy... .... .... .... .... ..... .... 37
4.1 Number and Locations of Fixed Points. .... .... .... ..... .... 40
4.2 Equidistribution . .... ..... .... .... .... .... .... ..... .... 53
4.3 Expanding Thurston Maps as Factors of the Left-Shift. ..... .... 73
5 Equilibrium States... .... ..... .... .... .... .... .... ..... .... 79
5.1 The Assumptions .... ..... .... .... .... .... .... ..... .... 83
5.2 Existence .. .... .... ..... .... .... .... .... .... ..... .... 84
5.3 Uniqueness. .... .... ..... .... .... .... .... .... ..... .... 104
5.4 Ergodic Properties ... ..... .... .... .... .... .... ..... .... 120
5.5 Co-homologous Potentials .. .... .... .... .... .... ..... .... 124
5.6 Equidistribution . .... ..... .... .... .... .... .... ..... .... 132
5.7 A Random Iteration Algorithm... .... .... .... .... ..... .... 134
ix
x Contents
6 Asymptotic h-Expansiveness.... .... .... .... .... .... ..... .... 137
6.1 Some Properties of Expanding Thurstons Maps.. .... ..... .... 139
6.2 Concepts from Graph Theory.... .... .... .... .... ..... .... 143
6.3 Proof of Theorem 6.1. ..... .... .... .... .... .... ..... .... 144
7 Large Deviation Principles..... .... .... .... .... .... ..... .... 161
7.1 Level-2 Large Deviation Principles.... .... .... .... ..... .... 164
7.2 Characterizations of the Pressure Pðf;/Þ ... .... .... ..... .... 166
7.3 Proof of Large Deviation Principles... .... .... .... ..... .... 169
7.4 Equidistribution Revisited... .... .... .... .... .... ..... .... 170
References.... .... .... .... ..... .... .... .... .... .... ..... .... 173
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 179
