Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Cover of the book
Tytuł książki

Topological and measurable dynamics of Lorenz maps

Seria
Rozprawy Matematyczne tom/nr w serii: 382 wydano: 1999
Zawartość
Warianty tytułu
Abstrakty
EN
Contents
1. Introduction.............................................................................................5
2. 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................................................................................30
3. 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.............................................................65
4. 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...........................................89
5. 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............................................................123
References..............................................................................................129
Index........................................................................................................132
EN
Acknowledgements
This 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".
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 382
Liczba stron
134
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXXII
Daty
wydano
1999
otrzymano
1998-11-25
poprawiono
1999-02-02
Twórcy
Bibliografia
  • [1] D. Berry and B. D. Mestel, Wandering intervals for Lorenz maps with bounded variation, Bull. London Math. Soc. 23 (1991) 183-189.
  • [2] A. M. Blokh and M. Y. Lyubich, Attractors of maps of the interval, in: Banach Center Publ. 23, PWN, Warszawa, 1989, 427-442.
  • [3] A. M. Blokh and M. Y. Lyubich, On decomposition of one-dimensional dynamical systems into ergodic components. The case of negative Schwarzian, Leningrad Math. J. 1 (1990), no. 1, 137-155.
  • [4] H. Bruin, Invariant measures of interval maps, PhD thesis, Technische Universiteit Delft, 1994.
  • [5] H. Bruin, Combinatorics of the kneading map, Internat. J. Bifurcation Chaos 5 (1995) 1339-1349.
  • [6] H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys. 168 (1995) 571-580.
  • [7] H. Bruin, Topological conditions for the existence of absorbing Cantor sets, Trans. Amer. Math. Soc. 350 (1998) 2229-2263.
  • [8] H. Bruin, G. Keller, T. Nowicki, and S. van Strien, Wild Cantor attractors exist, Ann. of Math. 143 (1996) 97-130.
  • [9] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Boston, 1980.
  • [10] P. Coullet, J.-M. Gambaudo et C. Tresser, Une nouvelle bifurcation de codimension 2: le collage de cycles, C. R. Acad. Sci. Paris Sér. I 299 (1984) 253-256.
  • [11] P. Coullet et C. Tresser, Itérations d'endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. I 287 (1978) 577-580.
  • [12] P. Coullet et C. Tresser, Itérations d'endomorphismes et groupe de renormalisation, J. Phys. C5 (1978) 25-28.
  • [13] W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. 25, Springer, Berlin, 1993.
  • [14] A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993) 263-297.
  • [15] J.-M. Gambaudo, Ordre, désordre, et frontière des systèmes Morse-Smale, thèse d'Etat, l'Université de Nice, 1987.
  • [16] J.-M. Gambaudo et al., New universal scenarios for the onset of chaos in Lorenz-type flows, Phys. Rev. Lett. 57 (1986) 925-928.
  • [17] J.-M. Gambaudo, P. A. Glendinning, and C. Tresser, Stable cycles with complicated structure, J. Phys. Lett. 46 (1985) 653-657.
  • [18] J.-M. Gambaudo, P. A. Glendinning, and C. Tresser, Collage de cycles et suites de Farey, C. R. Acad. Sci. Paris Sér. I 299 (1984) 711-714.
  • [19] J.-M. Gambaudo, O. Lanford III et C. Tresser, Dynamique symbolique des rotations, C. R. Acad. Sci. Paris Sér. I 299 (1984) 823-826.
  • [20] P. Glendinning and C. Sparrow, Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps, Phys. D 62 (1993) 22-50.
  • [21] J. Guckenheimer, A strange, strange attractor, in: J. E. Marsden and M. McCracken (eds), The Hopf Bifurcation Theorem and its Applications, Springer, 1976, 368-381.
  • [22] J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Publ. Math. I.H.E.S. 50 (1979) 307-320.
  • [23] F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math. 34 (1979) 213-237.
  • [24] F. Hofbauer, The topological entropy of the transformation x ↦ ax(1-x), Monatsh. Math. 90 (1980) 117-141.
  • [25] F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II, Israel J. Math. 38 (1981) 107-115.
  • [26] F. Hofbauer, The structure of piecewise monotonic transformations, Ergodic Theory Dynam. Systems 1 (1981) 159-178.
  • [27] F. Hofbauer, Monotonic mod one transformations, Studia Math. 80 (1984) 17-40.
  • [28] F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields 72 (1986) 359-386.
  • [29] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982) 119-140.
  • [30] F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990) 319-337.
  • [31] F. Hofbauer and G. Keller, Some remarks on recent results about S-unimodal maps, Ann. Inst. H. Poincaré 53 (1990) 413-425.
  • [32] F. Hofbauer and P. Raith, Topologically transitive subsets of piecewise monotonic maps, which contain no periodic points, Monatsh. Math. 107 (1989) 217-239.
  • [33] A. Homburg, Some global aspects of homoclinic bifurcations of vector fields, PhD thesis, Rijksuniversiteit Groningen, 1993.
  • [34] J. H. Hubbard and D. Schleicher, The spider algorithm, in: Proc. Sympos. Appl. Math. 49, Amer. Math. Soc., 1994, 155-180.
  • [35] J. H. Hubbard and C. T. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math. 43 (1990) 431-443.
  • [36] J. P. Keener, Chaotic behaviour in piecewise continuous difference equations, Trans. Amer. Math. Soc. 261 (1980) 589-604.
  • [37] G. Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989) 183-200.
  • [38] G. Keller, Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc. 314 (1989) 433-497.
  • [39] G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems 10 (1990) 717-744.
  • [40] G. Keller, Zeta functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys. 127 (1990) 459-477.
  • [41] G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992) 31-69.
  • [42] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math. 6, de Gruyter, Berlin, 1985.
  • [43] M. Lin, Mixing for Markov operators, Z. Wahrsch. Verw. Gebiete 19 (1971) 231-242.
  • [44] E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Sci. 20 (1963) 130-141.
  • [45] M. I. Malkin, On continuity of entropy of discontinuous mappings of the interval, Selecta Math. Sov. 8 (1989) 131-139.
  • [46] M. I. Malkin, Rotation intervals and the dynamics of Lorenz type mappings, Selecta Math. Sov. 10 (1991) 265-275.
  • [47] M. Martens and W. de Melo, Universal models for Lorenz maps, preprint, Oct. 1996.
  • [48] M. Martens, W. de Melo, P. Mendez, and S. van Strien, Cherry flows on the torus: towards a classification, Ergodic Theory Dynam. Systems 10 (1990) 531-553.
  • [49] W. S. Massey, A Basic Course in Algebraic Topology, Grad. Texts in Math. 127, Springer, Berlin, 1991.
  • [50] J. Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985) 177-195.
  • [51] J. Milnor and W. Thurston, On Iterated Maps of the Interval, Lecture Notes in Math. 1342, Springer, Berlin, 1988.
  • [52] M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Publ. Math. I.H.E.S. 53 (1981).
  • [53] W. Parry, The Lorenz attractor and a related population model, in: Lecture Notes in Math. 729, Springer, Berlin, 1979, 169-187.
  • [54] D. Rand, The topological classification of Lorenz attractors, Math. Proc. Cambridge Philos. Soc. 83 (1978) 451-460.
  • [55] D. Sands, Topological conditions for positive Lyapunov exponent in unimodal maps, PhD thesis, St. John's College, Cambridge, 1994.
  • [56] H. Thunberg, Some problems in unimodal dynamics, PhD thesis, KTH, Stockholm, 1996.
  • [57] C. Tresser, Nouveaux types de transitions vers une entropie topologique positive, C. R. Acad. Sci. Paris Sér. I 296 (1983) 729-732.
  • [58] M. Tsujii, A note on Milnor and Thurston's monotonicity theorem, in: Geometry and Analysis in Dynamical Systems (Kyoto, 1993), Adv. Ser. Dynam. Systems 14, World Sci., 1994, 60-62.
  • [59] M. Tsujii, A simple proof of monotonicity of entropy in quadratic family using Ruelle operator, preprint, October 1997.
  • [60] R. F. Williams, The structure of Lorenz attractors, Publ. Math. I.H.E.S. 50 (1979) 321-347.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 58F03; Secondary 58F11, 58F12, 58F14
Identyfikator YADDA
bwmeta1.element.zamlynska-bd295345-6842-4dfe-a29a-4ec72d7ba65d
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
Zawartość książki

rozwiń roczniki

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.