Conformal Dimension: Theory and Application PDF
Preview Conformal Dimension: Theory and Application
University L ECTURE Series Volume 54 Conformal Dimension Theory and Application John M. Mackay Jeremy T. Tyson American Mathematical Society Conformal Dimension Theory and Application University L ECTURE Series Volume 54 Conformal Dimension Theory and Application John M. Mackay Jeremy T. Tyson ERMIACAN MΑΓΕΩΜΕAΤTΡHΗEΤΟMΣ AΜTΗIΕΙΣΙΤΩCALSOYCTIE FOUNDED 1888 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Eric M. Friedlander (Chair) Benjamin Sudakov William P. Minicozzi II Tatiana Toro 2010 Mathematics Subject Classification. Primary 30L10, 28A78. For additional informationand updates on this book, visit www.ams.org/bookpages/ulect-54 Library of Congress Cataloging-in-Publication Data Mackay,JohnM.,1982– Conformaldimension: theoryandapplication/JohnM.Mackay,JeremyT.Tyson. p.cm. —(Universitylectureseries;v.54) Includesbibliographicalreferencesandindex. ISBN978-0-8218-5229-3(alk.paper) 1.Quasiconformalmappings. 2.Hausdorffmeasures. I.Tyson,JeremyT.,1972– II.Title. QA360.M285 2010 515(cid:2).93—dc22 2010014667 Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Requests for such permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by [email protected]. (cid:2)c 2010bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 151413121110 Dedicated to Pierre Pansu on the occasion of his fiftieth birthday Contents Preface ix List of Figures xiii Chapter 1. Background material 1 1.1. Notation and conventions 1 1.2. Mappings 1 1.3. Quasisymmetric invariants 8 1.4. Dimensions and measures 9 1.5. Notes 17 Chapter 2. Conformal gauges and conformal dimension 21 2.1. Conformal gauges 21 2.2. Conformal dimension 22 2.3. Notes 24 Chapter3. Gromovhyperbolicgroups and spacesand theirboundaries 25 3.1. The real hyperbolic plane and CAT(κ) spaces 25 3.2. Gromov hyperbolic metric spaces 26 3.3. Boundaries of rank one symmetric spaces 30 3.4. Gromov hyperbolic groups 40 3.5. Boundaries of hyperbolic buildings 43 3.6. Notes 45 Chapter 4. Lower bounds for conformal dimension 49 4.1. Rich curve families and lower bounds for conformal dimension 49 4.2. The Loewner condition and examples of spaces minimal for conformal dimension 55 4.3. The conformal dimension of the Sierpin´ski carpet 58 4.4. Annular LLC implies conformal dimension bounded away from one 61 4.5. Rich families of sets of conformal dimension one 62 4.6. Notes 64 Chapter 5. Sets and spaces of conformal dimension zero 67 5.1. Conformal dimension, porosity and uniform disconnectedness 67 5.2. Conformal dimension and unions 71 5.3. Conformal dimension and products 73 vii viii CONTENTS 5.4. Tukia’s example 77 5.5. Notes 78 Chapter 6. Gromov–Hausdorff tangent spaces and conformal dimension 79 6.1. Tangent spaces and lower bounds for conformal Assouad dimension 79 6.2. Lowering the conformal Assouad dimension 82 6.3. Notes 89 Chapter 7. Ahlfors regular conformal dimension 91 7.1. David–Semmes deformation theory 91 7.2. Cannon’s conjecture and Ahlfors regular conformal dimension 92 7.3. (cid:3)p cohomology of graphs 93 7.4. Dynamics of post-critically finite rational maps on S2 97 7.5. Notes 104 Chapter 8. Global quasiconformal dimension 107 8.1. Distortion of dimension by quasiconformal maps 107 8.2. Minimality for global quasiconformal dimension and middle interval Cantor sets 108 8.3. Global quasiconformal dimensions of self-similar sets 109 8.4. Conformal dimensions of self-affine sets 115 8.5. Sets with global quasiconformal dimension zero 122 8.6. Notes 125 Bibliography 131 Index 139 Preface Quasisymmetric maps are homeomorphisms that distort relative dis- tance in a uniform, scale-invariant fashion. Their role in geometric function theory is pervasive. Euclidean quasiconformal maps are locally quasisym- metric. Quasisymmetry provides a metric substitute for quasiconformality suitable for the study of maps defined on lower-dimensional submanifolds or subsets of an ambient Euclidean space. On the other hand, quasiconfor- malmapsofEuclideanspacesofsufficientlyhighdimension,orofsufficiently nicedomainstherein, arequasisymmetric. Itisbynowwellunderstoodthat quasisymmetry is a remarkably ubiquitous condition which also arises in many settings outside of classical function theory. For instance, the bound- ary extensions of quasi-isometries of hyperbolic groups are quasisymmetric. Quasisymmetric maps of the boundaries of strictly pseudoconvex domains in Cn, equipped with the standard Carnot-Carath´eodory metric adapted to their CR structures, control the behavior of proper holomorphic maps near the boundary. Julia sets of rational maps, parametrized by hyperbolic components of the moduli space, are quasisymmetrically equivalent. Other examples could be given. Determining the quasisymmetry type of a metric space is one of the premier open problems in geometric function theory. One of the most well- studied quasisymmetric invariants is conformal dimension. This invariant was introduced by Pansu in the late 1980’s as a tool for the quasi-isometric classification of the classical rank one symmetric spaces and lattices in such spaces. In recent years a significant body of mathematics has accumulated concerningconformaldimensionsofmetricspaceswhichmaynotnecessarily ariseasgroupboundaries. Otherapplicationsarebeginningtobeidentified, inareas as diverseas first-orderanalysisin metricspaces, dynamicsofratio- nal maps, Gromov hyperbolic geometry, and self-affine fractal geometry. In our view, the theory has developed and expanded to a point which justifies a presentation emphasizing its scope and maturity. We have attempted to provide such a presentation in the present survey. We provide a comprehensive picture of the current state of the art re- garding both computations and estimates of conformal dimension. Our pre- sentation emphasizes both pure theory—computation of the conformal di- mensions of metric spaces for its own sake, as an interesting exercise in geometric function theory—as well as applications of such computations in ix
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