Springer Monographs in Mathematics978-3-540-87525-3/1.pdf · Ghys, Penner, Rozzy, Sullivan and Thurston. Further, there are connections with the theorems of Arnold, Herman and Yoccoz

Springer Monographs in Mathematics

Alberto A. Pinto • David A. Rand •

Flávio Ferreira

Fine Structuresof HyperbolicDiffeomorphisms

Alberto A. Pinto David A. RandUniversity of Minho Mathematics InstituteDepartamento de Matemática (DM) University of WarwickCampus de Gualtar Coventry, CV4 7AL4710 - 057 Braga UKPortugal [email protected]@math.uminho.pt

Flávio FerreiraEscola Superior de Estudos Industriaise de GestãoInstituto Politécnico do PortoR. D. Sancho I, 9814480-876 Vila do [email protected]

ISBN 978-3-540-87524-6 e-ISBN 978-3-540-87525-3

DOI 10.1007/978-3-540-87525-3

Springer Monographs in Mathematics ISSN 1439-7382

Library of Congress Control Number: 2008935620

Mathematics Subject Classification (2000): 37A05, 37A20, 37A25, 37A35, 37C05, 37C15, 37C27,37C40, 37C70, 37C75, 37C85, 37E05, 37E05, 37E10, 37E15, 37E20, 37E25, 37E30, 37E45

c© 2009 Springer-Verlag Berlin Heidelberg

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springer.com

In celebration of the 60th birthday ofDavid A. Rand

For

Maria Guiomar dos Santos Adrego Pinto

Barbel Finkenstadt and the Rand kids: Ben, Tamsin, Rupert andCharlotte

Fernanda Amelia Ferreira and Flavio Andre Ferreira

Family and friends

Dedicated to Dennis Sullivan and Christopher Zeeman.

VI

Acknowledgments

Dennis Sullivan had numerous insightful discussions with us on thiswork. In particular, we discussed the construction of solenoid functions,train-tracks, self-renormalizable structures and pseudo-smooth structures forpseudo-Anosov diffeomorphisms.

We would like to acknowledge the invaluable help and encouragement offamily, friends and colleagues, especially Abdelrahim Mousa, Alby Fisher,Aldo Portela, Aloisio Araujo, Aragao de Carvalho, Athanasios Yannakopou-los, Baltazar de Castro, Barbel Finkenstadt, Bruno Oliveira, Carlos Matheus,Carlos Rocha, Charles Pugh, Dennis Sullivan, Diogo Pinheiro, Edson de Faria,Enrique Pujals, Etienne Ghys, Fernanda Ferreira, Filomena Loureiro, GabrielaGoes, Helena Ferreira, Henrique Oliveira, Hugo Sequeira, Humberto Mor-eira, Isabel Labouriau, Jacob Palis, Joana Pinto, Joana Torres, Joao Almeida,Joaquim Baiao, John Hubbard, Jorge Buescu, Jorge Costa, Jose Goncalves,Jose Martins, Krerley Oliveira, Lambros Boukas, Leandro Almeida, LeonelPias, Luciano Castro, Luis Magalhaes, Luisa Magalhaes, Marcelo Viana,Marco Martens, Maria Monteiro, Mark Pollicott, Marta Faias, Martin Peters,Mauricio Peixoto, Miguel Ferreira, Mikhail Lyubich, Nelson Amoedo, NicoStollenwerk, Nigel Burroughs, Nils Tongring, Nuno Azevedo, Pedro Lago, Pa-tricia Goncalves, Robert MacKay, Rosa Esteves, Rui Goncalves, Saber Elaydi,Sebastian van Strien, Sofia Barros, Sofia Cerqueira, Sousa Ramos, StefanoLuzzatto, Stelios Xanthopolous, Telmo Parreira, Vilton Pinheiro, WarwickTucker, Welington de Melo, Yunping Jiang and Zaqueu Coelho.

We thank IHES, CUNY, SUNY, IMPA, the University of Warwick andthe University of Sao Paulo for their hospitality. We also thank Calouste Gul-benkian Foundation, PRODYN-ESF, Programs POCTI and POCI by FCTand Ministerio da Ciencia e da Tecnologia, CIM, Escola de Ciencias da Uni-versidade do Minho, Escola Superior de Estudos Industriais e de Gestao doInstituto Politecnico do Porto, Faculdade de Ciencias da Universidade doPorto, Centros de Matematica da Universidade do Minho e da Universidade doPorto, the Wolfson Foundation and the UK Engineering and Physical SciencesResearch Council for their financial support. We thank the Golden Medal dis-tinction of the Town Hall of Espinho in Portugal to Alberto A. Pinto.

Alberto PintoDavid Rand

Flavio Ferreira

Preface

The study of hyperbolic systems is a core theme of modern dynamics. Onsurfaces the theory of the fine scale structure of hyperbolic invariant sets andtheir measures can be described in a very complete and elegant way, and isthe subject of this book, largely self-contained, rigorously and clearly written.It covers the most important aspects of the subject and is based on severalscientific works of the leading research workers in this field.

This book fills a gap in the literature of dynamics. We highly recommendit for any Ph.D student interested in this area. The authors are well-knownexperts in smooth dynamical systems and ergodic theory.

Now we give a more detailed description of the contents:Chapter 1. The Introduction is a description of the main concepts in hyper-

bolic dynamics that are used throughout the book. These are due to Bowen,Hirsch, Mane, Palis, Pugh, Ruelle, Shub, Sinai, Smale and others. Stable andunstable manifolds are shown to be Cr foliated. This result is very useful in anumber of contexts. The existence of smooth orthogonal charts is also proved.This chapter includes proofs of extensions to hyperbolic diffeomorphisms ofsome results of Mane for Anosov maps.

Chapter 2. All the smooth conjugacy classes of a given topological modelare classified using Pinto’s and Rand’s HR structures. The affine structures ofGhys and Sullivan on stable and unstable leaves of Anosov diffeomorphismsare generalized.

Chapter 3. A pair of stable and unstable solenoid functions is associatedto each HR structure. These pairs form a moduli space with good topologi-cal properties which are easily described. The scaling and solenoid functionsintroduced by Cui, Feigenbaum, Fisher, Gardiner, Jiang, Pinto, Quas, Randand Sullivan, give a deeper understanding of the smooth structures of one andtwo dimensional dynamical systems.

Chapter 4. The concept of self-renormalizable structures is introduced.With this concept one can prove an equivalence between two-dimensional hy-perbolic sets and pairs of one-dimensional dynamical systems that are renor-malizable (see also Chapter 12). Two C1+ hyperbolic diffeomorphisms that

VIII Preface

are smoothly conjugate at a point are shown to be smoothly conjugate. Thisextends some results of de Faria and Sullivan from one-dimensional dynamicsto two-dimensional dynamics.

Chapter 5. A rigidity result is proved: if the holonomies are smooth enough,then the hyperbolic diffeomorphism is smoothly conjugate to an affine model.This chapter extends to hyperbolic diffeomorphisms some of the results ofAvez, Flaminio, Ghys, Hurder and Katok for Anosov diffeomorphisms.

Chapter 6. An elementary proof is given for the existence and uniqueness ofGibbs states for Holder weight systems following pioneering works of Bowen,Paterson, Ruelle, Sinai and Sullivan.

Chapter 7. The measure scaling functions that correspond to the Gibbsmeasure potentials are introduced.

Chapter refsmeasures. Measure solenoid and measure ratio functions areintroduced. They determine which Gibbs measures are realizable by C1+ hy-perbolic diffeomorphisms and by C1+ self-renormalizable structures.

Chapter 9. The cocycle-gap pairs that allow the construction of all C1+

hyperbolic diffeomorphisms realizing a Gibbs measure are introduced.Chapter 10. A geometric measure for hyperbolic dynamical systems is

defined. The explicit construction of all hyperbolic diffeomorphisms with sucha geometric measure is described, using the cocycle-gap pairs. The results ofthis chapter are related to Cawley’s cohomology classes on the torus.

Chapter 11. An eigenvalue formula for hyperbolic sets on surfaces withan invariant measure absolutely continuous with respect to the Hausdorffmeasure is proved. This extends to hyperbolic diffeomorphisms the Livsic-Sinai eigenvalue formula for Anosov diffeomorphisms preserving a measureabsolutely continuous with respect to Lebesgue measure. Also given here isan extension to hyperbolic diffeomorphisms of the results of De la Llave, Marcoand Moriyon on the eigenvalues for Anosov diffeomorphisms.

Chapter 12. A one-to-one correspondence is established between C1+ arcexchange systems that are C1+ fixed points of renormalization and C1+ hyper-bolic diffeomorphisms that admit an invariant measure absolutely continuouswith respect to the Hausdorff measure. This chapter is related to the work ofGhys, Penner, Rozzy, Sullivan and Thurston. Further, there are connectionswith the theorems of Arnold, Herman and Yoccoz on the rigidity of circlediffeomorphisms and Denjoy’s Theorem. These connections are similar to theones between Harrison’s conjecture and the investigations of Kra, Norton andSchmeling.

Chapter 13. Pinto’s golden tilings of the real line are constructed (seePinto’s and Sullivan’s d-adic tilings of the real line in the Appendix C). Thesegolden tilings are in one-to-one correspondence with smooth conjugacy classesof golden diffeomorphisms of the circle that are fixed points of renormalization,and also with smooth conjugacy classes of Anosov diffeomorphisms with an in-variant measure absolutely continuous with respect to the Lebesgue measure.The observation of Ghys and Sullivan that Anosov diffeomorphisms on the

Preface IX

torus determine circle diffeomorphisms having an associated renormalizationoperator is used.

Chapter 14. Thurston’s pseudo-Anosov affine maps appear as periodicpoints of the geodesic Teichmuller flow. The works of Masur, Penner, Thurstonand Veech show a strong link between affine interval exchange maps andpseudo-Anosov affine maps. Pinto’s and Rand’s pseudo-smooth structuresnear the singularities are constructed so that the pseudo-Anosov maps aresmooth and have the property that the stable and unstable foliations areuniformly contracted and expanded by the pseudo-Anosov dynamics. Classi-cal results for hyperbolic dynamics such as Bochi-Mane and Viana’s dualityextend to these pseudo-smooth structures. Blow-ups of these pseudo-Anosovdiffeomorphisms are related to Pujals’ non-uniformly hyperbolic diffeomor-phisms.

Appendices. Various concepts and results of Pinto, Rand and Sullivan forone-dimensional dynamics are extended to two-dimensions. Ratio and cross-ratio distortions for diffeomorphisms of the real line are discussed, in the spiritof de Melo and van Strien’s book.

Rio de Janeiro, Brazil Jacob PalisJuly 2008 Enrique R. Pujals

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Stable and unstable leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Marking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Interval notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Basic holonomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Foliated atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Foliated atlas Aι(g, ρ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Straightened graph-like charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 Orthogonal atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.10 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 HR structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Conjugacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 HR - Holder ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Foliated atlas A(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 HR Orthogonal atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Complete invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.7 Moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.8 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Realized solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Holder continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Matching condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6 Cylinder-gap condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.7 Solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.8 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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4 Self-renormalizable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Train-tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Markov maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Exchange pseudo-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5 Markings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.6 Self-renormalizable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.7 Hyperbolic diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.8 Explosion of smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.9 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.1 Complete sets of holonomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 C1,1 diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 C1,HDι

and cross-ratio distortions for ratio functions . . . . . . . . . 595.4 Fundamental Rigidity Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.5 Existence of affine models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.6 Proof of the hyperbolic and Anosov rigidity . . . . . . . . . . . . . . . . . 675.7 Twin leaves for codimension 1 attractors . . . . . . . . . . . . . . . . . . . 685.8 Non-existence of affine models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.9 Non-existence of uniformly C1,HDι

complete sets ofholonomies for codimension 1 attractors . . . . . . . . . . . . . . . . . . . . 71

5.10 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.1 Dual symbolic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Weighted scaling function and Jacobian . . . . . . . . . . . . . . . . . . . . 746.3 Weighted ratio structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4 Gibbs measure and its dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7 Measure scaling functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1 Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 Extended measure scaling function . . . . . . . . . . . . . . . . . . . . . . . . 867.3 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 Measure solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.1 Measure solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.1.1 Cylinder-cylinder condition . . . . . . . . . . . . . . . . . . . . . . . . . 948.2 Measure ratio functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.3 Natural geometric measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.4 Measure ratio functions and self-renormalizable structures . . . . 998.5 Dual measure ratio function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048.6 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Contents XIII

9 Cocycle-gap pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.1 Measure-length ratio cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.2 Gap ratio function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.3 Ratio functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.4 Cocycle-gap pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.5 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

10 Hausdorff realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11910.1 One-dimensional realizations of Gibbs measures . . . . . . . . . . . . . 11910.2 Two-dimensional realizations of Gibbs measures . . . . . . . . . . . . . 12210.3 Invariant Hausdorff measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

10.3.1 Moduli space SOLι . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13110.3.2 Moduli space of cocycle-gap pairs . . . . . . . . . . . . . . . . . . . 13210.3.3 δι-bounded solenoid equivalence class of Gibbs measures132

10.4 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

11 Extended Livsic-Sinai eigenvalue formula . . . . . . . . . . . . . . . . . . 13511.1 Extending the eigenvalues’s result of De la Llave, Marco and

Moriyon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13511.2 Extending the eigenvalue formula of A. N. Livsic and Ja. G.

Sinai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14011.3 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

12 Arc exchange systems and renormalization . . . . . . . . . . . . . . . . 14312.1 Arc exchange systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

12.1.1 Induced arc exchange systems . . . . . . . . . . . . . . . . . . . . . . . 14512.2 Renormalization of arc exchange systems . . . . . . . . . . . . . . . . . . . 148

12.2.1 Renormalization of induced arc exchange systems . . . . . 15012.3 Markov maps versus renormalization . . . . . . . . . . . . . . . . . . . . . . . 15212.4 C1+H flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15512.5 C1,HD rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15612.6 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

13 Golden tilings (in collaboration with J.P. Almeida andA. Portela) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16113.1 Golden difeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

13.1.1 Golden train-track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16213.1.2 Golden arc exchange systems . . . . . . . . . . . . . . . . . . . . . . . 16313.1.3 Golden renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16513.1.4 Golden Markov maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

13.2 Anosov diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16813.2.1 Golden diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16913.2.2 Arc exchange system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17013.2.3 Markov maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17213.2.4 Exchange pseudo-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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13.2.5 Self-renormalizable structures . . . . . . . . . . . . . . . . . . . . . . . 17413.3 HR structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17413.4 Fibonacci decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

13.4.1 Matching condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17613.4.2 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17613.4.3 The exponentially fast Fibonacci repetitive property . . . 17713.4.4 Golden tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17713.4.5 Golden tilings versus solenoid functions . . . . . . . . . . . . . . 17813.4.6 Golden tilings versus Anosov diffeomorphisms . . . . . . . . . 181

13.5 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

14 Pseudo-Anosov diffeomorphisms in pseudo-surfaces . . . . . . . . 18314.1 Affine pseudo-Anosov maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18314.2 Paper models Σk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18414.3 Pseudo-linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18614.4 Pseudo-differentiable maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

14.4.1 Cr pseudo-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19414.4.2 Pseudo-tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19514.4.3 Pseudo-inner product on Σk . . . . . . . . . . . . . . . . . . . . . . . . 195

14.5 Cr foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19814.6 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A Appendix A: Classifying C1+ structures on the real line . . . 201A.1 The grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201A.2 Cross-ratio distortion of grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202A.3 Quasisymmetric homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 204A.4 Horizontal and vertical translations of ratio distortions . . . . . . . 207A.5 Uniformly asymptotically affine (uaa) homeomorphisms . . . . . . 214A.6 C1+r diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224A.7 C2+r diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228A.8 Cross-ratio distortion and smoothness . . . . . . . . . . . . . . . . . . . . . . 232A.9 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

B Appendix B: Classifying C1+ structures on Cantor sets . . . . 235B.1 Smooth structures on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

B.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236B.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239B.3 (1 + α)-contact equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

B.3.1 (1 + α) scale and contact equivalence . . . . . . . . . . . . . . . . 241B.3.2 A refinement of the equivalence property . . . . . . . . . . . . . 242B.3.3 The map Lt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243B.3.4 The definition of the contact and gap maps . . . . . . . . . . . 246B.3.5 The map Ln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247B.3.6 The sequence of maps Ln converge . . . . . . . . . . . . . . . . . . 247B.3.7 The map L∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

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B.3.8 Sufficient condition for C1+α−-equivalent . . . . . . . . . . . . . 252

B.3.9 Necessary condition for C1+α−-equivalent . . . . . . . . . . . . 252

B.4 Smooth structures with α-controlled geometry and boundedgeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254B.4.1 Bounded geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

B.5 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

C Appendix C: Expanding dynamics of the circle . . . . . . . . . . . . 261C.1 C1+Holder structures U for the expanding circle map E . . . . . . . 261C.2 Solenoids (E,S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263C.3 Solenoid functions s : C → R

+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265C.4 d-Adic tilings and grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267C.5 Solenoidal charts for the C1+Holder expanding circle map E . . . 269C.6 Smooth properties of solenoidal charts . . . . . . . . . . . . . . . . . . . . . 271C.7 A Teichmuller space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272C.8 Sullivan’s solenoidal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273C.9 (Uaa) structures U for the expanding circle map E . . . . . . . . . . 274C.10 Regularities of the solenoidal charts . . . . . . . . . . . . . . . . . . . . . . . . 275C.11 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

D Appendix D: Markov maps on train-tracks . . . . . . . . . . . . . . . . 279D.1 Cookie-cutters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279D.2 Pronged singularities in pseudo-Anosov maps . . . . . . . . . . . . . . . 280D.3 Train-tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

D.3.1 Train-track obtained by glueing . . . . . . . . . . . . . . . . . . . . . 282D.4 Markov maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283D.5 The scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

D.5.1 A Holder scaling function without a correspondingsmooth Markov map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

D.6 Smoothness of Markov maps and geometry of the cylinderstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291D.6.1 Solenoid set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291D.6.2 Pre-solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292D.6.3 The solenoid property of a cylinder structure . . . . . . . . . 293D.6.4 The solenoid equivalence between cylinder structures . . . 295

D.7 Solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297D.7.1 Turntable condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298D.7.2 Matching condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

D.8 Examples of solenoid functions for Markov maps . . . . . . . . . . . . 299D.8.1 The horocycle maps and the diffeomorphisms of the

circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300D.8.2 Connections of a smooth Markov map. . . . . . . . . . . . . . . . 301

D.9 α-solenoid functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302D.10 Canonical set C of charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303D.11 One-to-one correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

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D.12 Existence of eigenvalues for (uaa) Markov maps . . . . . . . . . . . . . 307D.13 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

E Appendix E: Explosion of smoothness for Markov families . 313E.1 Markov families on train-tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

E.1.1 Train-tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313E.1.2 Markov families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314E.1.3 (Uaa) Markov families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315E.1.4 Bounded Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

E.2 (Uaa) conjugacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319E.3 Canonical charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324E.4 Smooth bounds for Cr Markov families . . . . . . . . . . . . . . . . . . . . . 325

E.4.1 Arzela-Ascoli Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330E.5 Smooth conjugacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331E.6 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347