Real and Complex Dynamical Systems edited by Bodil Branner and Poul Hjorth Mathematical Institute, The Technical University of Denmark, Lyngby, Denmark
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Series C: Mathematical and Physical Sciences - Vol. 464 Proceedings of the NATO Advanced Study Institute on Real and Complex Dynamical Systems Hiller/IJd, Denmark June 2o-July 2, 1993
A C.I.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-90-481-4565-2 ISBN 978-94-015-8439-5 (eBook) DOI 10.1007/978-94-015-8439-5
AII Rights Reserved @ 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint of the hardcover 1st edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, includ ing photo- copying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS
Preface vii
List of participants ix
Viviane BALADI: Dynamical Zeta Functions 1
Chris BUDD: The Global Dynamics of Impact Oscillators ...... 27
Chris BUDD: Grazing in Impact Oscillators ...... 47
Adrien DOUADY: Topological Entropy of Unimodal Maps ...... 65
John H. HUBBARD and Ralph W. OBERSTE-VORTH: H enon Mappings in the Complex Domain ...... 89
Bruce KITCHENS: Symbolic Dynamics, Group Automorphisms and Markov Partitions 133
Silvina P. DAWSON, Roza GALEEVA, John MILNOR and Charles TRESSER: A Monotonicity Conjecture for Real Cubic Maps 165
Colin SPARROW: Dynamics of Ordinary Differential Equations 185
Sebastian van STRIEN: Real Bounds in Complex Dynamics 211
Marcelo VIANA: Homoclinic Bifurcations and Strange Attractors 231
Jean-Christophe YOCCOZ: Introduction to Hyperbolic Dynamics 265
Lai-Sang YOUNG: Ergodic Theory of Differentiable Dynamical Systems 293
Index ...... 337 PREFACE
This volume contains edited versions of 11 contributions given by main speakers at the NATO Advanced Study Institute on lReal and Complex Dynamical Systems in Hiller0d, Denmark, June 20th - July 2nd, 1993.
The vision of the institute was to illustrate the interplay between two important fields of Mathematics: Real Dynamical Systems and Complex Dynamical Systems. The interaction between these two fields has been growing over the years. Problems in Real Dynamical Systems have recently been solved using complex tools in the real or by extension to the complex. In return, problems in Complex Dynamical Systems have been settled using results from Real Dynamical Systems. The programme of the institute was to examine the state of the art of central parts of both Real and Complex Dynamical Systems, to reinforce contact between the two aspects of the theory and to make recent progress in each accessible to a larger group of mathematicians.
We wish to express our sincere thanks to all lecturers and participants for having helped to make this ASI a success. We acknowledge the extensive amount of time the invited main speakers had to spend to prepare the expository lectures, and to prepare their manuscripts for publication. Special thanks go to Adrien Douady, Se- bastian van Strien and Lai-Sang Young for their efforts as members of the Scientific Organizing Committee; moreover, to the conference secretaries Lone Aagesen and Tove Densted for their competent management of all matters great & small during the conference; to the very professional staff at the Apotekerforeningens Kursuse- jendom (where the Institute was held); and to two local participants, Pia Willumsen and Dan S0rensen for assistance above and beyond the call of mere participation. We would also like to thank Jacob Dylander and acknowledge his patient and carefull work of formatting and assembling the manuscripts of this book.
The ASI was funded principally by NATO, with additional support from MIDIT (Modelling, Non-linear Dynamics and Irreversible Thermodynamics) at The Tech- nical University of Denmark, the Danish Natural Science Research Council, the Carlsberg Foundation, the Thomas B. Thrige Foundation, the Otto B. M0nsted Foundation, the Mathematical Institute at The Technical University of Denmark and the Danish Mathematical Society. We would like to thank all these organiza- tions for their support. For its efforts on behalf of this ASI we are most grateful to the Scientific Affairs Division of NATO, particularly to Dr. Luis V. da Cunha, the Director of the ASI programme.
Bodil Branner, Scientific Director Poul Hjorth
vii LIST OF PARTICIPANTS
Simonette ABENDA Viviane BALADI S.I.S.S.A. Laboratoire de Mathematiques Via Beiruth, 4 Ecole Normale Superieure de Lyon 1-34014 Trieste 46, Allee d'Italie ITALY F -69364 Lyon Cedex 17 e-mail abenda ix x Jean-Yves BRIEND Amy CHIU Ecole Normale Superieure de Lyon Department of Mathematics 46, Allee d'Italie Boston University F -69364 Lyon Cedex 07 111, Cummington St. FRANCE Boston, MA 02215 e-mail ens-Iyon.:fr UNITED STATES e-mail olympiadIDmath. bu. edu Henk BRUIN Delft University of Technology Adrien DOUADY TWI General Mathematics Universite de Paris-Sud Mekelweg 4 Departement de Mathematiques NL-2600 GA Delft Batiment 425 NETHERLANDS F-91405 Orsay e-mail FRANCE e-mail adrien.douadylDens.fr Morten BR0NS Mathematical Institute Jakob DYLANDER The Technical University of Denmark Mathematical Institute Building 303 University of Copenhagen DK-2800 Lyngby Universitetsparken 5 DENMARK DK-2100 Copenhagen 0 e-mail bronsCOmat. dtu. elk DENMARK e-mail dylanderCOmath.ku.elk Chris BUDD School of Mathematics Adam EPSTEIN University of Bristol Department of Mathematics University Walk Cal. Tech. GB-Bristol, BS8 1TW Pasadena , CA UNITED KINGDOM UNITED STATES e-mail . uk e-mail elagolDcunyvm . bitnet Xavier BUFF Juan Francisco ESTRADA GARCIA Ecole Normale Superieure Universite de Paris-Sud 45 rue d'Ulm Departement de Mathematiques F-75005 Paris Cedex Batiment 425 FRANCE F-91405 Orsay Cedex e-mail bu:f:fIDens. ens.:fr FRANCE Manuel CHAVES Nuria FAGELLA Grupo de Matematica Pura Department of Mathematics Faculdade de Ciencias Boston University Praca Gomes Teixeira 111, Cummington St. P-4000 Porto Boston, MA 02215 PORTUGAL UNITED STATES e-mail machavesID:fc.up.pt e-mail nuria FANG Jinqing Coby GEYSEL China Institute of Atomic Energy University of Amsterdam P.O. Box 275-27 Faculty of Math.F and Computer Sci- Beijing 102413 ence P. R. of China Plantage Muidergracht 24 NL-1018 TV Amsterdam NETHERLANDS Jerome FEHRENBACH e-mail cobyCDfwLuva.nl Ecole Normale Superieure 45 rue d'Ulm F -75005 Paris Cedex Aleksei Antonovich GLUTSUK FRANCE Department of Mathematics e-mail fehrenbaCDclipper.ens.fr Moscow State University Leninskie Gori 117234 Moscow Bj0rn FELSAGER RUSSIA Kildegard Gymnasium Kildegardsvej 87 Kari HAG DK-2900 Hellerup Department of Mathematics DENMARK The Norwegian Institute of Technology e-mail felsagerCDmat.dtu.dk N- 7034 Trondheim NORWAY Marguerite FLEXOR e-mail kariCDimf.unit.no U niversite de Paris-Sud Departement de Mathematiques Peter HAISSINSKY Biitiment 425 Ecole Normale Superieure de Cachan F-91405 Orsay Cedex 61, Avenue du President Wilson FRANCE F -94230 Cachan Cedex e-mail flexorCDmatups.matups.fr FRANCE Roza GALEEVA Ernst HANSEN Department of Mathematics Mathematical Institute UWM P.O. Box 413 University of Copenhagen Milwaukee, WI 53201 U niversitetsparken 5 UNITED STATES DK-2100 Copenhagen 0 e-mail rozaCD DENMARK archimedes.math.uwm.edu e-mail erhansenCDmath.ku.dk Daniele GERARD Kai HANSEN U niversite de Paris-Sud Niels Bohr Institute Departement de Mathematiques University of Copenhagen Biitiment 425 Blegdamsvej 15-17 F-91405 Orsay Cedex DK-2100 Copenhagen 0 FRANCE DENMARK e-mail dgerardCDmatups.matups.fr e-mail khansenCDnbi vax. nbi xii Jane HAWKINS Pascal HUBERT Math. Dept. CB#3250 Ecole Normale Superieure University of North Carolina at Chapel 46 Allee d'Italie Hill F -69364 Lyon Cedex Chapel Hill, NC 27514 FRANCE UNITED STATES e-mail hubertClens.ens-lyon.fr e-mail jmhawkGunc . bitnet Axel HUNDEMER Sandra HAYES Matematisches Institut Mathematisches Institut der der Technischen Universitat Munchen Technischen Universitat Munchen Arcisstrasse 21 Arcisstrasse 21 D-8000 Munchen 2 D-8000 Munchen GERMANY GERMANY e-mail hayesCl mathematik.tu-muenchen.de Arne JAKOBSEN Department of Mathematics Florent HIVERT The Norwegian Institute of Technology Ecole Normale Superieure N-7034 Trondheim 45 rue d'Ulm NORWAY F -75005 Paris Cedex e-mail arnejaClimf.unit.no FRANCE e-mail hivertClens.ens.fr Emmanuelle JEANDENANS Universite de Bourgogne Poul HJORTH Departement de Mathematiques Mathematical Institute Lab. de Topologie - URA 755, B.P. 138 The Technical University of Denmark F-21004 Dijon Cedex Building 303 FRANCE DK-2800 Lyngby e-mail topologCl DENMARK satie.u-bourgogne.fr e-mail hjorthClmat.dtu.dk Habib JELLOULI Jun HU Universite Math. Dept. of CUNY de Paris-Sud Graduate Center Departement de Mathematiques 33, W 42nd Street Batiment 425 New York, NY 10036 F-91405 Orsay Cedex UNITED STATES FRANCE e-mail jellouliClmatups.matups.fr e-mail huj Clcunyvmsl. gc . cuny. edu John H. Hubbard Yunping JIANG Department of Mathematics Mathematics Department White Hall Queens College of CUNY Cornell University 65-30 Kissena Bid. Ithaca, NY 14853 NY 11367 UNITED STATES UNITED STATES e-mail hubbardClmath. cornell. edu e-mail j iangClmath. sunysb. edu xiii Peter JONES Carsten KNUDSEN Department of Mathematics Physics Department Yale University The Technical University of Denmark New Haven Building 309 CT 06520 DK-2S00 Lyngby UNITED STATES DENMARK e-mail carsten Mattias JONSSON Hartje KRIETE Matematiska Inst. 115 Lehrstuhl II fur Mathematik Kungl. Tekniska Hogskolan RWTH Aachen S-10044 Stockholm D-52056 Aachen SWEDEN GERMANY e-mail mjo Jeremy KAHN Department of Mathematics Harbir LAMBA University of California School of Mathematics Berkeley, CA 94720 University of Bristol UNITED STATES University Walk e-mail kahnC!math. berkeley. edu GB-Bristol B58 1TW UNITED KINGDOM e-mail h.lambaC!bristol.ac . uk Bruce KITCHENS Mathematical Sciences Department J ens Christian LARSEN IBM T.J. Watson Research Center Mathematical Institute Yorktown Heights The Technical University of Denmark New York 10598 Building 303 UNITED STATES DK-2S00 Lyngby e-mail kitchC!watson.ibm.com DENMARK e-mail maijcl Oliver KNILL Laurent LE FLOCH Mathematik Departement Laboratoire de geometrie analytique ETH Zentrum UFR de Mathematiques CH-S092 Zurich Universite de Rennes I SWITZERLAND F -35042 Rennes Cedex e-mail knill LI Weigu Fernando Jorge Soares MOREIRA Department of Mathematics Faculdade de Ciencias Moscow State University Grupo de Matematica Pura Leninskie Gori Praca Gomes Teixeira 117234 Moscow P-4000 Porto RUSSIA PORTUGAL e-mail fsmoreir Bahtiyar Ozgiir SARIOGLU Colin T. SPARROW Bilkent University Department of Mathematics and Math- Department of Mathematics ematical Statistics Bilkent 06533 Ankara 16, Mill Lane TURKEY GB-Cambridge CB2 1SB e-mail sozgurtlltrbilun. bitnet UNITED KINGDOM e-mail c. t. sparrowtll statslab.cam.ac.uk R. Phil SCHAFER The University of Texas at Austin Sebastian VAN STRIEN Department of Mathematics University of Amsterdam Austin, Texas 78712-1082 Faculty of Math. and Computer Sci- UNITED STATES ence e-mail philiptllmath. utexas. edu Plantage Muidergracht 24 NL-1018 TV Amsterdam Dierk SCHLEICHER NETHERLANDS Department of Mathematics e-mail strienClfwi.uva.nl Cornell University Dan S0RENSEN Ithaca, NY 14853 Mathematical Institute UNITED STATES The Technical University of Denmark e-mail dierktllmath. cornell. edu Building 303 DK-2800 Lyngby Pierrette SENTENAC DENMARK Universite de Paris-Sud e-mail dantllmat.dtu.dk Departement de Mathematiques, Bat. 425 TAN Lei F-91405 Orsay Cedex Ecole Normale Superieure de Lyon FRANCE 46, Allee d'Italie F -69364 Lyon Cedex 07 e-mail sentenactllmatups.matups.fr FRANCE e-mail tanleitllfrens161. bitnet Mitsuhiro SHlSHlKURA Department of Mathematics Hans THUNBERG Tokyo Institute of Technology Matematiska Inst. 115 Ohokayama, Meguro Kungl. Tekniska Hogskolan Tokyo 152 S-10044 Stockholm JAPAN SWEDEN e-mail mitsutllmath.titech.ac.jp e-mail hassettllmath.kth.se Shigehiro USHIKI Jan SKRZYPCZAK Graduate School of Human and Envi- Institute of Mathematics ronmental Studies Warsaw University Kyoto University ul. Banacha 2 606-01 Kyoto PL-02-097 Warsaw JAPAN POLAND e-mail ushikitll e-mail janskrztllmimuw.edu.pl platon.kula.kyoto-u.ac.jp xvii Marcelo VIANA Pia WILLUMSEN Departamento de Matematica Pura Mathematical Institute Faculdade de Ciencias do Porto The Technical University of Denmark P-4000 Porto Building 303 PORTUGAL DK-2800 Lyngby and DENMARK Instituto de Matematica Pura e Apli- e-mail piabat.dtu.dk cad a Est. D. Castorina 110 22460 Rio de Janeiro WUHe Brasil Math Group e-mail viana«limpa.br Int. Centre for Theoretical Physics Fabienne VUILLEMIN P.O. Box 586 Universite de Bourgogne 1-34100 Trieste Departement de Mathematiques ITALY Laboratoire de Topologie - URA 755 e-mail wuhe«lictp.trieste.it B.P. 138 F-21004 Dijon Cedex FRANCE Jean-Christophe YOCCOZ e-mail topolog«lfrccubl1. bitnet Universite de Paris-Sud Departement de Mathematiques Marysia T. WEISS Batiment 425 Hofstra University F-914050rsay Hempstead, New York 11550 FRANCE UNITED STATES e-mail matmtw«lhofstra. bitnet Anne Marie WILKINSON Lai-Sang YOUNG Department of Mathematics Department of Mathematics University of California University of California Berkeley, CA 94720 Los Angeles, CA 90024 UNITED STATES UNITED STATES e-mail wilkinso A Axiom A a priori bounds 222 attractor 325, 327, 328 absolute continuity diffeomorphism ..... 9 of a foliation . 327 of the WS-foliation 331 B absolutely continuous baker transformation 294, 314 conditional measures on basic set ...... 286 unstable manifolds315, 328, 331 basin of attraction 253, 263 invariant measure 255, 321 Bernoulli shift 294, 297, 328, 332 accesible boundary . . 121 Bernoulli transformation 295 accounting scheme 190, 205 bifurcation aCim ...... 321, 323 border-collision . . . 52 adapted norm 268 discontinuous grazing . 35 admissible solutions 198 gluing . . 198, 203, 207 amalgamation 139 grazing ...... 48 analytic map homoclinic . . . . 194, 232, 233 hyperbolic . 16 associated to a horseshoe 247 anti-monotonicity 193 Shil'nikov 208 approximate Hopf 188 contractive directions 261 period-doubling 243 critical set . . . 261 . U4 critical values. . 261 period-multiplying 206 area-conservative case 242, 252 saddle-node 243 area-dissipativeness 252, 253, 260 billiard balls . 27 attraction billiards . . 301 basin of . · . 253, 263 bimodal map 166 attractor ... · ... 324 Birkhoff Ergodic Theorem 294 Axiom A . 325, 327, 328 block map ...... 134 periodic . . . 242 "bones" ...... 173 strange · . 253, 254 border-collision bifurcation . 52 of Henon type 254, 262, 263 bound period ...... 256 prevalence of 262 bound time automorphism total 257 toral 146 bounded non-linearity 223 automorphism of subshift bounded type 212 of finite type . 145 Branner 213 autonomous system 186 averaging C method of .. 188 canonical system of conditional probability measures . 298 337 338 INDEX Cantor set critically finite map ... 72 thickness of a 250 cross-ratio . . . . 215, 216 set of curves 237 cross-ratio distortion . 217 CaratModory loop .76 crossed mapping 94, 96 cascade degree of . .97 period-doubling 192, 193, 197 cubic maps .. . 165 of period-doubling bifurcations244 D Cech cohomology . . 121 decomposition center manifold . . . 244 spectral 284 chain recurrence 276, 284 degree of Hopf bifurcation . . . . 188 invariant set 268 horizontal-like . . . . . 99 isentrope . 168 horseshoe 188, 236, 314 itinerary .. 238 homo clinic bifurcation J associated to a 247 Jakobson's Theorem. . .. 324 Hu .75 Jiang ...... 75 Hubbard ..... 213 Jordan form away from zero 141 hyperbolic analytic map · 16 Julia set ...... 17,95 hyperbolic automorphism 266 hyperbolic continuation 270, 274 K hyperbolic distance . 216 Keller . 213, 223 hyperbolic map . . . . . 266 kneading . · 172 hyperbolic polynomial . . 102 angle 66,79 hyperbolic set 236, 247 invariant · .66 continuation of 239 matrix. · .23 thick 251 theory . 192, 196, 208 hyperbolicity 266 Kobayashi metric .98 generic 177 Koebe Lemma . 215 Koebe Principle 218 I Krein-Milman Theorem 320 impact .27 L grazing .47 Lakes of Wada . 92, 125 low velocity · 51 Lebesgue measure 214 impact map P .29 positive . 214 impact oscillator .27 Levin .... 213 impact side .49 limit capacity inductive trick 222 248 limit point . . 289 infinite stretching · 31 limit set . . . infinite-dimensional space 289 line of tangencies of homeomorphisms . 114 247, 248 local connectivity infinitely renormalizable . 211 · . 213 local ergodicity . infinite-to-one factor map 143 · . 330 local product structure initial conditions 152, 282-284 local stable manifold . sensitive dependence on 253,255 281, 310, 326 local stable set . . . . intermittent behaviour . .48 · . 152 local unstable manifold interval map 310, 326 local unstable set . 152 piecewise monotone · 18 locally maximal. . . . intrinsic differentiability 251 · . 283 Lorenz equations . . . invariant cone 301, 333 185, 187,208 Lorenz geometric model invariant cone field 236 196 low velocity impact invariant foliation 239 · .... 51 Lyapunov chart . . regularity of . 239 · ... 306 Lyapunov exponent invariant measure . 294 299, 314, 333 Lyubich .. absolutely continuous 255 321 · ... 212 ergodic decomposition of. ' 321 M for attractors 324 magic word · . . . 143 space of ...... 320 manifold INDEX 341 center . 244 n-precritical point .69 stable . 107,277 n-shift global 282 full .... 134 local 281 nuclear operator · 14 strong-stable 244 a unstable .. 107 obstacle maps with singularities 313 rigid .28 Markov one-dimensional .48 condition 159 operator extension . 21 nuclear · 14 matrix .. .72 Perron-Frobenius 323 partition. 72, 157, 162 transfer 2, 323 Marstrand's theorem 250 orbit following 190 measure order type . 173 dimension of . . 317 oscillator Hausdorff dimension of 317 impact . .27 measure preserving transformation 294 Oseledec's Theorem . 299 meromorphic extension 5, 8, 9, 14, 19, over-Markov packing .72 22 meromorphic function 11, 15 p method of averaging . 188 packing metric entropy 295 over-Markov .72 Milnor ...... 66 parabolic point . .75 Misiurewicz point .75 parametrized families mixing of diffeomorphisms 232 exponential .11 partition topological . 135 Markov .72 transformation 295 period-doubling modulus of an annulus 99, 216 bifurcation 243 monotone cascade 192, 193, 197, 244 pIeceWIse 167 periodic attractor 242 monotonicity . 165 periodic behaviour 255 of entropy map .85 periodic orbit . 186 mpt ...... 294 primitive ... · 1 periodic point 289 N period-multiplying bifurcation 206 negative Schwarzian derivative .22 Perron value ...... 136 n-itinerary ...... 67 Perron-Frobenius non wandering operator ...... 323 point 290 theorems ...... 3 set 290 persistence of homo clinic tangencies251 non-uniformly expanding map 323 Pesin's formula 314-316 non-impact side...... 49 Petersen . . 221 norm phase (Pi . . .30 adapted ...... 268 phase space 186 normally contracting saddle-node 244 I{) Nowicki ...... 213, 223 degree of . 143 342 INDEX piecewise monotone interval map 18 resonances . . . . .11 piecewise-monotone map 167 resonant frequency .34 plateau .86 restitution Poincare metric 95, 215 coefficient of .28 polynomial return ..... 256 quadratic .74 reversible system 201, 206 polynomial tuning · 81 Riemann surface laminations · 93 pre-loaded . . · 36 Riemann zeta function · 10 pressure rigid obstacle . . . · 28 topological ...... · 4 rigidity result. . . 213 prevalence of strange attractors 262 Ruelle's inequality 314, 315 prime-orbit theorem . . 10 S primitive periodic orbit · 1 product structure saddle-node 244 local 282-284 normally contracting 244 projective limit · 90 bifurcation . 243 pseudo-orbit . 275 cycle critical 244-246, 262 p-telescope . . · 95 sawtooth Pustylnikov map · 27 stunted 170 scaling laws 225 Q Schwarzian derivative 217 Q-orbits ...... 198, 200 negative . . . . . 22 quadratic family 240, 243, 245 Schwarz's Lemma. 98, 215, 217, 218 quadratic polynomial ..... 74 sectional dissipativeness 242, 252 quadratic tangency 232, 234, 241 sensitive dependence quadratic-like on initial conditions 253, 255 family .. 240, 252, 254, 258 Shadowing Lemma 275 map ...... 252 Shannon-Breiman-McMillan quasi-conformal Theorem. 297 homeomorphism 221 shift map ...... 115 Bernoulli. 294, 297, 328, 332 R suspension of . · 7 ramified covering lamination 117 shift equivalence problem 141 rational . . . . · 3 shift map ...... 238 rational function · 9 Shil'nikov homo clinic bifurcation 208 real bounds 215 Shil'nikov homoclinic orbit . 194 rectangle. 161 0" A ...... · 3 recurrence . 289 Sinai-Ruelle-Bowen measure 315, 318, chain 276, 284 324, 328, 331-333 regularity of Sinai-Ruelle-Bowen measure .11 invariant foliation 239 singular map the stable and unstable stable and unstable sets for. 245 foliations...... 247 singularity renormalization 18, 67, 240, 242, 244, square-root. . . .52 251 size of U' in U .99 renormalization theory . 211 smallest interval trick 220 repellor ...... · 17 solenoid ...... 120, 150 INDEX 343 2-adic . . 150 subdivision. . . . . · 69 generalized 151 critical · 69 Space Ball . . . 28 subshift of finite type 3, 134 space of invariant measures . 320 automorphism of 145 spectral decomposition 284 Sullivan ...... 212 spectral radius Sullivan's renormalization result 220 essential . . . . . 4 super resonant . . .35 splitting algorithm 259, 261 suspensions of shifts · 7 square-root singularity . . . 52 Swi<}tek . . . .. 213 SRB measure. 315, 318, 324, 328, symbolic dynamics 238 s-rectangle . . .. 237 T stability tangency dynamical 239, 246 homoclinic 232, 234, 235 stable and unstable sets persistence of . .... 251 for singular maps . 245 line of .. . . 247,248 stable and unstable foliation quadratic 232, 234, 241 regularity of . . . . . 247 telescopes . .94 stable and unstable manifolds tent map ... 66,73 for non uniformly tesselation . . .71 hyperbolic systems . . 311 thick hyperbolic set 251 stable disk . . 100, 112 thickness of a Cantor set 250 stable manifold 107,312 Thurston ...... 66 global . . 282, 312, 326 topological conjugacy 152, 238 local 281, 310, 326 topological entropy . 3, 65, 156, 165 Stable Manifold Theorem 277 topological invariant . 191, 208 for a Fixed Point 309 topological pressure . ... 4 stable set 152 topological transitivity . 295, 326 local 152 topologically critical. . .69 Standard family topologically conservative 252 mixing homeomorphism 291 standard map 302 topologically state splitting 138 mixing transformation . 135 Steenrod homology 121 topologically sticking region · 30 transitive transformation . 135 strange attractor . 253, 254 toral automorphism 146 of Henon type 254, 262, 263 total bound time 257 prevalence of 262 trajectory .29 stretching transfer operator 2, 323 infinite · 31 transition rule 134, 150, 159 strong-stable foliation 245 transitive strong-stable manifold 244 doubly...... 136, 152 stunted sawtooth . . 170 transitive homeomorphism 291 sub resonant . . . . · 35 transitive transformation . .. 152 Subadditive Ergodic Theorem 303 transitivity...... 238 subadditive sequence .67 topological . . . . 135, 295, 326 transverse homo clinic point 344 INDEX tuning ...... 67, 80, 86 by blowing up .80 by modification · 81 polynomial · 81 2-adic solenoid 150 U unfolding generic ...... 232, 241 uniformly hyperbolic ... . 325 uniformly hyperbolic attractor 326 unimodal map . 73 unstable disk ...... 100 unstable manifold. . .. 107, 312 absolutely continuous conditional measures on 315 , 328, 331 global . 312, 326 local 310, 326 unstable set 152 local 152 u-rectangle . 236 V variational principle 4, 65 W wandering domains ...... 108 weighted dynamical zeta function . 1 wild at tractors . . 212 existence of 214 Williams conjecture 141 word magIc . . . . 143 W' -foliation absolute continuity of 331 y Yoccoz 75, 213 Yoccoz puzzle . . 213 Z zeta function 136, 149 dynamical · 1 weighted · 1 Riemann. · 10