Topological and measurable dynamics of Lorenz maps

St. Pierre Matthias

GDML_Books, (1999), p.

Résumé

Contents1. Introduction.............................................................................................52. Markov extensions.................................................................................17  2.1. Lorenz maps.....................................................................................17  2.2. The Hofbauer tower..........................................................................18  2.3. The extended Hofbauer tower...........................................................24  2.4. The decomposition of the Markov diagram.......................................25  2.5. Renormalization................................................................................303. Hopf decompositions and attractors.......................................................37  3.1. Transfer operators............................................................................37  3.2. The Hopf decomposition...................................................................41  3.3. The asymptotic behaviour of points on the tower..............................48  3.4. Wandering intervals..........................................................................51  3.5. Attractors and invariant measures....................................................56  3.6. Shadowing the critical orbits.............................................................654. Kneading theory....................................................................................70  4.1. The kneading invariant.....................................................................70  4.2. The splitting of itineraries.................................................................75  4.3. Admissibility conditions.....................................................................77  4.4. Renormalization from a combinatorial viewpoint...............................86  4.5. Rotation numbers and rotation intervals...........................................895. Families of Lorenz maps........................................................................96  5.1. The Thurston algorithm....................................................................97  5.2. Parameter dependence of the kneading invariant..........................103  5.3. The gluing bifurcation.....................................................................107  5.4. Homoclinic bifurcation points..........................................................110  5.5. Monotonic Lorenz families..............................................................115  5.6. Proof of the Full Family Theorem...................................................120  5.7. The quadratic Lorenz family............................................................123References..............................................................................................129Index........................................................................................................132

AcknowledgementsThis paper is based on the author's doctoral thesis written at the Institute of Mathematics of the Friedrich-Alexander-University Erlangen-Nürnberg under the supervision of Professor Dr. Gerhard Keller. I would like to thank him very much for proposing this interesting subject to me and for all the support he gave me during the making of this thesis. He always had an open ear for my questions and the remarkable capability of tracking down the vital points of my problems extremely quick and then giving me many helpful suggestions and good new ideas.During his stay in Erlangen, Henk Bruin was an invaluable source of knowledge for me. In particular, I learned many details about the combinatorics of Hofbauer towers from him. He was a very careful reader of early and late versions of this manuscript and made many useful comments and remarks, for which I am very grateful.I would also like to thank Sebastian van Strien for his hospitality during my visit to the University of Warwick, which was very motivating and encouraging for me. He showed me the striking beauty and simplicity of the Thurston algorithm, which inspired me to write the program that produced many of the figures included in this work.The work on this thesis was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the Schwerpunktprogramm "Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme".

1991 Mathematics Subject Classification: Primary 58F03; Secondary 58F11, 58F12, 58F14

Détail

Zbl 1211.37048

EUDML-ID : urn:eudml:doc:271747