Chapter 8 One-dimensional maps

Handbook of Dynamical Systems

Volumn 1, Issue PART A, 2002, Pages 599-664

  Jakobson, M ^a   Swiatek G ^b  

^a University of Maryland   (United States)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67649361716     PISSN: 1874575X     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-575X(02)80010-8     Document Type: Article

References (94)

1. Franks J., and Misiurewicz M. Topological methods in dynamics. Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 547-598
2. Hasselblatt B. Hyperbolic dynamical systems. Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 239-319
3. Hasselblatt B., and Katok A. Principal structures. Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 1-203
4. Lind D., and Schmidt K. Symbolic and algebraic dynamical systems. Handbook of Dynamical Systems Vol. 1A (2002), Elsevier, Amsterdam 765-812
5. Alsedá L.I., Llibre J., and Misiurewicz M. Combinatorial Dynamics and Entropy in Dimension One (1993), World Scientific Publishing Company, River Edge, NJ
6. Carleson L., and Gamelin T. Complex Dynamics (1993), Springer-Verlag, New York
7. Collet P., and Eckmann J.-P. Iterated Maps on an Interval as Dynamical Systems (1980), Birkhäuser, Basel
8. Graczyk J., and Świa{ogonek}tek G. The Real Fatou Conjecture. Ann. of Math. Studies Vol. 144 (1998), Princeton University Press, Princeton, NJ
9. Katok A., and Hasselblatt B. Modern Theory of Dynamical Systems (1995), Cambridge University Press, Cambridge
10. McMullen C. Renormalization and 3-Manifolds which Fiber over the Circle. Annals of Math. Studies Vol. 142 (1996), Princeton University Press, Princeton, NJ
11. De Melo W., and Van Strien S. One-Dimensional Dynamics (1993), Springer-Verlag, New York
12. Milnor J., and Thurston W. On Iterated Maps of the Interval. Lecture Notes in Math. Vol. 1342 (1988), Springer- Verlag, Berlin
13. Sharkovsky A., Kolyada S., Sivak A., and Fedorenko V. Dynamics of One-Dimensional Maps (1997), Kluwer Academic Publishers, Dordrecht
14. Yoccoz J.-C. Petits diviseurs en dimension 1. Astérisque Vol. 231 (1995)
15. Adler R.L. F-expansions revisited. Lecture Notes in Math. Vol. 318 (1973), Springer-Verlag, Berlin 1-5
16. Alekseev V.M. Symbolic dynamics (1976), 11th Math. School, Publications of the Math. Inst. of Ukrainian Acad. Sci.
17. 2 on (-1,1). Ann. Math. 122 (1985) 1-25
18. Blokh A., and Lyubich M. Attractors of maps of the interval. Dynamical Systems and Ergodic Theory. Banach Center Publications Vol. 23 (1989) 427-442
19. Blokh A., and Misiurewicz M. Collet-Eckmann maps are unstable. Comm. Math. Phys. 191 (1998) 61-70
20. Blokh A., and Misiurewicz M. Rotating an interval and a circle. Trans. Amer. Math. Soc. 351 1 (1999) 63-78
21. Bruin H. Combinatorics of the kneading map. Internat. J. Bifurcations and Chaos 5 (1995) 1339-1349
22. Bruin H. Topological conditions for the existence of invariant measures for unimodal maps. Ergodic Theory Dynamical Systems 59 (1994) 433-451
23. Bruin H., Keller G., Nowicki T., and van Strien S. Wild Cantor attractors exist. Ann. of Math. 143 (1996) 97-130
24. Denker M., Przytycki F., and Urbanski M. On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory Dynamical Systems 16 (1996) 255-266
25. Graczyk J. Harmonic scaling for smooth families of diffeomorphisms of the circle. Nonlinearity 4 (1991) 935-959
26. Graczyk J., and Smirnov S. Collet, Eckmann, and Hölder. Invent. Math. 133 (1998) 69-96
27. Graczyk J., and Świa{ogonek}tek G. Critical circle maps near bifurcation. Comm. Math. Phys. 176 (1996) 227-260
28. Graczyk J., and Świa{ogonek}tek G. Singular measures in circle dynamics. Comm. Math. Phys. 157 (1993) 157-230
29. Guckenheimer J. Sensitive dependence on initial conditions for one dimensional maps. Comm. Math. Phys. 70 (1979) 133-160
30. Guckenheimer J., and Johnson S. Distortion of S-unimodal maps. Ann. Math. 132 (1990) 71-130
31. ∞ Denjoy counterexample. Ergodic Theory Dynamical Systems 1 (1981) 261-272
32. Herman M. Are there critical points on the boundaries of Siegel disks?. Comm. Math. Phys. 99 (1985) 593-612
33. M. Herman, Mésure de Lebesgue et nombre de rotation. Proc. Symp. Geometry and Topology, Lect. Notes in Math., Vol. 597, Springer-Verlag, Berlin, 271-293.
34. Hofbauer F. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34 (1979) 213-237
35. Hofbauer F., and Keller G. Some remarks about recent results on S-unimodal maps. Ann. Inst. de Henri Poincarè, Physique Théorique 53 (1990) 413-425
36. 23-13. Hotbauer F., and Keller G. Equilibrium states for piecewise monotonic transformations. Ergodic Theory Dynamical Systems 2 (1982)
37. Hotbauer F., and Keller G. Quadratic maps without asymptotic measure. Comm. Math. Phys. 127 (1990) 319-337
38. Hu J., and Sullivan D. Topological conjugacy of circle diffeomorphisms. Ergodic Theory Dynamical Systems 17 (1997) 173-186
39. M. Jakobson, On smooth mappings of the circle into itself, Math. Sbor. 14. 161-185.
40. 1. Jakobson M.V. Universal behavior and stochasticity for one-dimensional dynamical systems. Proceedings of 1st World Congress of the Bernoulli Society. VNU Sc. Press, Utrecht (1987) 85-90
41. Jakobson M. Quasisymmetric conjugacies for some one-dimensional maps inducing expansion. Contem- porary Math. 135 (1992) 203-211
42. Jakobson M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981) 39-88
43. Jakobson M. Topological and metric properties of one-dimensional endomorphisms. Sov. Math. Dokl. 19 (1978) 1452-1455
44. Jakobson M., and Świa{ogonek}tek G. Metric properties of non-renormalizable S-unimodal maps. Ergodic Theory Dynamical Systems 14 (1994) 721-755
45. Jakobson M., and Świa{ogonek}tek G. Metric properties of non-renormalizable S-unimodal maps: II. Quasisymmetric conjugacy classes. Ergodic Theory Dynamical Systems 15 (1995) 871-938
46. S. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys. 110, 185-190.
47. Jonker L., and Rand D. Bifurcations in one dimension. I. The nonwandering set. Invent. Math. 62 (1981) 347-365
48. Katznelson Y., and Ornstein D. The absolute continuity of the conjugation of certain diffeomorphisms of the circle. Ergodic Theory Dynamical Systems 9 (1989) 681-691
49. Katznelson Y., and Ornstein D. The differentiability of conjugation of certain diffeomorphisms of the circle. Ergodic Theory Dynamical Systems 9 (1989) 643-680
50. Keller G. Markov extensions, zeta-functions and Fredholm theory for piecewise-invertible dynamical systems. Trans. Amer. Math. Soc. 314 (1989) 433-497
51. first version 1987, Preprint. Keller G. Exponents, attractors and Hopf decompositions for interval maps. Ergodic Theory Dynamical Systems 10 (1990) 717-744
52. see also. Kozlovski O. Structural stability in one-dimensional dynamics. Ph.D. thesis (1998), University of Amsterdam
53. Kozlovski O. Getting rid of the negative Schwarzian condition. Ann. Math. 152 (2000) 743-762
54. Ledrappier F. Some properties of absolutely continuous invariant measures on an interval. Ergodic Theory Dynamical Systems 1 (1981) 77-93
55. Ledrappier F., and Misiurewicz M. Dimension of invariant measures for maps with exponent zero. Ergodic Theory Dynamical Systems 5 (1985) 595-610
56. Levin G., and Van Strien S. Local connectivity of the Julia sets of real polynomials. Ann. of Math. 147 (1998) 471-541
57. Lyubich M. Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. 140 (1994) 347-404
58. Lyubich M., and Milnor J. The dynamics of the Fibonacci polynomial. J. Amer. Math. Soc. 6 (1993) 425-457
59. Manning A. A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergodic Theory Dynamical Systems 1 (1981) 451-459
60. Mañé R. Hyperbolicity, sinks and measure in one dimensional dynamics. Comm. Math. Phys. 100 (1985) 495-524
61. Mañé R. Erratum. Comm. Math. Phys. 112 (1987) 721-724
62. R. Mañé, P. Sad, D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. 16, 193-217.
63. M. Martens, W. De Melo S. Van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 273-318
64. McCluskey H., and Manning A. Hausdorff dimension for horseshoes. Ergodic Theory Dynamical Systems 3 (1983) 251-260
65. Milnor J. On the concept of attractor. Comm. Math. Phys. 99 (1985) 177-195
66. Misiurewicz M. Structure of mappings of an interval with zero entropy. Publ. Math. IHES 53 (1981) 5-17
67. Misiurewicz M. Absolutely continuous measures for certain maps of an interval. Publ. Math. IHES 53 (1981) 17-51
68. Misiurewicz M., and Nitecki Z. Combinatorial patterns for maps of the interval. Mem. Amer. Math. Soc. 94 (1991) 456
69. Misiurewicz M., and Szlenk W. Entropy of piecewise monotone mappings. Studia Math. 67 (1980) 45-63
70. Nowicki T., and Przytycki F. The conjugacy of Collet-Eckmann's map of the interval with the tent map is Hölder continuous. Ergodic Theory Dynamical Systems 9 (1989) 379-388
71. Nowicki T., and Przytycki F. Topological invariance of the Collet-Eckmann property for S-unimodal maps. Fund. Math. 155 (1998) 33-43
72. Nowicki T., and Sands D. Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132 (1998) 633-680
73. Nowicki T., and Van Strien S. Absolutely continuous invariant measures under the summability condition. Invent. Math. 105 (1991) 123-136
74. Parry W., and Pollicott M. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque (1990) 187-188
75. Peréz-Marco R. Fixed points and circle maps. Acta Math. 179 (1997) 243-294
76. Przytycki F. Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials. Trans. Amer. Math. Soc. 350 (1998) 717-742
77. Przytycki F., and Rohde S. Porosity of Collet-Eckmann Julia sets. Fund. Math. 155 (1998) 189-199
78. V.A. Rokhlin Exact endomorphisms of Lebesgue space, Izv. Akad. Nauk SSSR, Ser. Math. 25, 499-530
79. Ruelle D. Repellers for real analytic maps. Ergodic Theory Dynamical Systems 2 (1982) 99-107
80. Rychlik M. Bounded variation and invariant measures. Studia Math. 76 1 (1983) 69-80
81. Rychlik M. Another proof of Jakobson's theorem and related results. Ergodic Theory Dynamical Systems 8 (1988) 83-109
82. Sharkovski A.N. Coexistence of cycles of a continuous map of a line into itself. Ukrainian Math. J. 16 (1964) 61-71
83. Addendum to the book. Sinai Ya.G. Some rigorous results on decay of correlations. In: Zaslavski G.M. (Ed). Statistical Irreversibility in Nonlinear Systems (1970), Nauka, Moscow 124-139
84. Slodkowski Z. Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991) 347-355
85. Sullivan D. Quasiconformal homeomorphisms and dynamics I: a solution of Fatou-Julia problem on wandering domains. Ann. Math. 122 (1985) 401-418
86. Sullivan D. Bounds, quadratic differentials and renormalization conjectures. American Mathematical Society Centennial Publications Vol. 2 (1991), American Mathematical Society, Providence, RI
87. Świa{ogonek}tek G. On critical circle homeomorphisms. Boll. Soc. Math. Bras. 29 (1998) 329-351
88. Świa{ogonek}tek G. One-dimensional maps and Poincaré metric. Nonlinearity 4 (1992) 81-108
89. Thieullen Ph., Tresser C., and Young L.-S. Positive Lyapunov exponent for generic one-parameter families of unimodal maps. J. Anal. Math. 64 (1994) 121-172
90. Tsujii M. Positive Lyapunov exponents in families of one-dimensional dynamical systems. Invent. Math. 111 (1993) 113-137
91. Vul E.B., Sinai Ya.G., and Khanin K.M. Feigenbaum universality and thermodynamic formalism. Russian Math. Surveys 39 3 (1984) 1-30
92. Walters P. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1975) 937-971
93. Yoccoz J.-C. Conjugaison des difféomorphismes analytiques du cercle, Manuscript. See also the same author's: An introduction to small divisors problems. In: Waldschmidt M. (Ed). From Number Theory to Physics (1992), Springer-Verlag, Berlin 659-679
94. 184-21. Zygmund A. Smooth functions. In: Hulanicki A., et al. (Ed). Selected Papers of Antoni Zygmund Vol. 2 (1989), Kluwer Academic Publishers, Dordrecht