Smooth unimodal maps in the 1990s

Ergodic Theory and Dynamical Systems

Volumn 19, Issue 2, 1999, Pages 263-287

  Graczyk, Jacek ^a   Świa̧tek, Grzegorz ^b  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0040912687     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S014338579914286X     Document Type: Review

References (86)

1. V. Baladi and M. Viana. Strong stochastic stability and rate of mixing for unimodal maps. Ann. Sci. Éc. Norm. Sup. 4 (1996), 483-517.
2. 2 on (-1, 1). Ann. Math. 122 (1985), 1-25.
3. M. Benedicks and L. Carleson. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73-169.
4. M. Benedicks and L.-S. Young. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergod. Th. & Dynam. Sys. 12 (1992), 13-37.
5. A. Blokh and M. Lyubich. Attractors of maps of the interval. In Dynamical Systems and Ergodic Theory. Banach Center Publications, Vol. 23, 1989, pp. 427-442; see also [BLy2].
6. A. Blokh and M. Lyubich. Measurable dynamics of S-unimodal maps of the interval. Ann. Sci. Éc. Norm. Sup. 24 (1991), 545-573.
7. A. Blokh and M. Misiurewicz. Collet-Eckmann maps are unstable. Commun. Math. Phys. 191 (1988), 61-70.
8. B. Branner and J. H. Hubbard. The iteration of cubic polynomials. Part II: patterns and parapatterns. Acta Math. 169 (1992), 229-325.
9. H. Bruin. Invariant measures of interval maps. PhD Thesis, Technical University of Delft, 1994.
10. H. Bruin. Combinatorics of the kneading map. Int. J. Bifurcation Chaos 5 (1995), 1339-1349.
11. H. Bruin, G. Keller, T. Nowicki and S. van Strien. Wild Cantor attractors exist. Ann. Math. 143 (1996), 97-130; submitted in 1994.
12. L. Carleson, P. Jones and J.-C. Yoccoz. Julia and John. Bol. Soc. Bras. Mat. 25 (1994), 1-30.
13. P. Collet and J.-P. Eckmann. Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1983), 13-46.
14. S. P. Dawson, R. Galeeva, J. Milnor and C. Tresser. Monotonicity conjecture for real cubic maps. In NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci. 464. Kluwer, Dordrecht, 1995.
15. M. Denker, F. Przytycki and M. Urbanski. On the transfer operator for rational functions on the Riemann sphere. Ergod. Th. & Dynam. Sys. 16 (1996), 255-266.
16. A. Douady and J. H. Hubbard. On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Sup. (Paris) 18 (1985), 287-343
17. E. De Faria. Asymptotic rigidity of scaling ratios for critical circle mappings. Preprint, Stony Brook, 1996. Ergod. Th. & Dynam. Sys. to appear.
18. E. De Faria and W. De Melo. Rigidity of critical circle mappings, Parts I and II. Preprints, Stony Brook, 1997.
19. J. Graczyk. Dynamics of circle maps with flat interval. Part of 1990 PhD Thesis. Preprint, Queen's University at Kingston, 1991. Fund. Math. to appear.
20. J. Graczyk and S. Smirnov. Collet, Eckmann, & Hölder. Invent. Math. 133 (1988), 69-96.
21. J. Graczyk and G. S̀wia̧tek. Singular measures in circle dynamics. Commun. Math. Phys. 157 (1993), 157-230.
22. J. Graczyk and G. S̀wia̧tek. Generic hyperbolicity in the logistic family. Ann. Math. 146 (1997), 1-52; submitted in August 1992.
23. J. Graczyk and G. Świa̧tek. Induced expansion for quadratic polynomials. Ann. Sci. Éc. Norm. Sup. 29 (1996), 399-482; first version June 1993.
24. J. Graczyk G. Świa̧tek. Polynomial-like property for real quadratic polynomials. Topology Proc. 21 (1996), 33-112; first version January 1995.
25. J. Graczyk and G. Świa̧tek. The real Fatou conjecture. Ann. of Math. Stud. 144, Princeton University Press, Princeton, NJ, 1998.
26. J. Guckenheimer. Sensitive dependence on initial conditions for one dimensional maps. Commun. Math. Phys. 70 (1979), 133-160.
27. J. Guckenheimer. Limit sets of S-unimodal maps with zero entropy. Commun. Math. Phys. 110 (1987), 655-659.
28. J. Guckenheimer and S. Johnson. Distortion of S-unimodal maps. Ann. Math. 132 (1990), 71-130.
29. Ch. Heckman. Monotonicity and the construction of quasiconformal conjugacies in the real cubic family. Thesis, SUNY, Stony Brook, 1996.
30. B. Hinkle. Parabolic limits of renormalization. Preprint, Stony Brook, 1997.
31. F. Hofbauer. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34 (1979), 213-237.
32. F. Hofbauer and G. Keller. Quadratic maps without asymptotic measure. Commun. Math. Phys. 127 (1990), 318-337.
33. F. Hofbauer and G. Keller. Some remarks about recent results on S-unimodal maps. Ann. Inst. de Henri Poincaré, Physique Théorique 53 (1990), 413-425.
34. M. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81 (1981), 39-88.
35. M. Jakobson. Quasisymmetric conjugacies for some one-dimensional maps inducing expansion. Contemporary Mathematics 135 (1992), 203-211; first version 1989, manuscript.
36. M. Jacobson and G. Świa̧tek. Metric properties of non-renormalizable S-unimodal maps. Ergod. Th. & Dynam. Sys. 14 (1994), 721-755; first version 1991, preprint IHES.
37. M. Jakobson and G. Świa̧tek. Metric properties of non-renormalizable S-unimodal maps: II. Quasisymmetric conjugacy classes. Ergod. Th. & Dynam. Sys. 15 (1995), 871-938; first version September 1991, Stony Brook preprint.
38. S. Johnson. Singular measures without restriction intervals. Commun. Math. Phys. 110 (1987), 185-190.
39. G. Keller. Markov extensions, zeta-functions and Fredhom theory for piecewise-invertible dynamic systems. Trans. Amer. Math. Soc. 314 (1989), 433-497.
40. G. Keller. Exponents, attractors and Hopt decompositions for intervals maps. Ergod. Th. & Dynam. Sys. 10 (1990), 717-744.
41. G. Keller and T. Nowicki. Spectral theory, zeta functions, and the distribution of periodec orbits for Collect-Eckmann maps. Commun. Math. Phys. 149 (1992), 31-69.
42. G. Keller and T. Nowicki. Fibonacci maps re(al) visited. Ergod. Th. & Dynam. Sys. 15 (1995), 97-130.
43. O. Kozlovski. Structural stability in one-dimensional dynamics. PhD Thesis, University of Amsterdam, 1998.
44. O.E. Lanford. A computer-assisted proof of the Feigenbaum conjecture. Bull. Amer. Math. Soc. 6 (1982), 427-434.
45. G. Levin and S. van Strien. Local connectivity of the Julia set of real polynomials. Ann. Math.147 (1998), 471-541; first version December 1994, manuscript.
46. R. de la Llave and P. Schafer. Rigidity properties of one dimensional expanding maps and applications to renormalization. Preprint, 1996.
47. M. Lyubich. Dynamics of complex polynomials, Part I-II. Acta Math. 178 (1997), 185-297; first version October 1993, Stony Brook preprint.
48. M. Lyubich and M. Yampolsky. Dynamics of quadratic polynomials: complex bounds for real maps. Ann. Inst. Fourier (Grenoble) 47 (1997), 1219-1255; first version April 1996, preprint MSRI.
49. M. Lyubich. Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math. 140 (1994), 347-404; first version November 1992, stony Brook preprint.
50. M. Lyubich. Dynamics of quadrstic maps, Part III. Preprint, Stony Brook, 1996.
51. M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint, Stony Brook, 1997.
52. M. Lyubich and J. Milnor. The dynamics of the Fibonacci polynomial. J. Amer. Math. Soc. 6 (1993), 425-457; first version 1991, Stony Brook preprint
53. R. Mañé. On a theorem of Fatou. Bol. Soc. Bras. Mat. 24 (1993), 1-12.
54. R. Mañé, P. Sad and D. Sullivan. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Sup. 16 (1983), 193-217.
55. M. Martens. The periodic points of renormalization. Ann. Math. 147 (1998), 543-584.
56. M. Martens. Distortion results and invariant Cantor sets of unimodal mappings. Ergod. Th. & Dynam. Sys. 14 (1994), 331-349; based on the 1990 PhD thesis, University of Delft.
57. M. Martens and T. Nowicki. Invariant measures for typical quadratic maps. Preprint, Stony Brook, 1996.
58. W. de melo and S. van Strine. One-Dimensional Dynamics. Springer, New York, 1993.
59. W. de Melo and S. van Strien. A structure theorem in one-dimensional dynamics. Ann. Math. 129 (1989), 519-546.
60. C. McMullen. Renormalization ans 3-manifolds which fiber over the circle. Ann. Math. Studies 142. Princetone University Press, Princeton, NJ, 1996.
61. J. Milnor. On the concept of attractor. Commun. Math. Phys. 99 (1985), 177-195.
62. J. Milnor. The Yoccoz theorem on local connectivity of Julia sets. A proof with pictures. Class Notes, Stony Brook, 1991-1992.
63. J. Milnor and W. Thurston. On Iterated Maps of an Interval (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 465-563.
64. T. Nowicki. Symmetric S-unimodal mappings and positive Lyapunov exponents. Ergod. Th. & Dynam. Sys. 5 (1985), 611-616.
65. T. Nowicki and F. Przytycki. The conjugacy of Collet-Eckmann's map of the interval with the tent map is Hölder continuous. Ergod. Th. & Dynam. Sys. 9 (1989), 379-388.
66. T. Nowicki and F. Przytycki. Topological invariance of the Collet-Eckmann property for S-unimodal maps. Fund. Math. 155 (1998), 33-43.
67. T. Nowicki and D. Sands. Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132 (1998), 633-680.
68. T. Nowicki and S. van Strien. Absolutely continuous invariant measures under the summability condition. Invent. Math. 105 (1991), 123-136.
69. F. Przytycki. Iterations of holomorphic Collet-Eckmann Maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials. Trans. Amer. Math. Soc. 350 (1998), 717-742.
70. F. Przytycki and S. Rohde. Porosity of Collet-Eckmann Julia sets. Fund. Math. 155 (1998), 189-199.
71. M. Rychlik. Another proof of Jakobson's theorem and related results. Ergod. Th. & Dynam. Sys. 8 (1988), 93-109.
72. M. Rychlik and E. Sorets. Regularity and other properties of absolutely invariant measures for certain maps for the quadratic family. Commun. Math. Phys. 150 (1992), 217-236.
73. D. Sands. Topological conditions for positive Lyapunov exponent in Unimodal maps. PhD Thesis, Cambridge University, 1994, pp. 1-107.
74. D. Sands. Complex bounds. Manuscript, October 2, 1996.
75. Z. Slodkowski. Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991), 347-355.
76. S. van Strien Smooth dynamics on the interval (with an emphasis on quadratic-like maps). New Directions in Differentiable Dynamical Systems (London Math. Soc. Lecture Note Ser., 127). Cambridge University Press, 1985, pp. 57-119.
77. D. Sullivan. Quasiconformal homeomorphisms and dynamics I: a solution of Fatou-Julia problem on wandering domains. Ann. Math. 122 (1985), 401-418.
78. D. Sullivan Bounds, quadratic differentials and renormalization conjetures. In American Mathematical Society Centennial Publications, Vol. 2. American Mathematical Society, Providence, RI, 1991, pp. 417-466.
79. G. S̀wia̧tek. Rational rotation numbers for maps of the circle. Commun. Math. Phys. 119 (1988), 109-128.
80. F. Tangerman and J. Veerman. Scalings in circle maps I. Commun. Math. Phys. 134 (1990), 89-107.
81. P. Thieullen, C. Tresser, and L.-S. Young. Exposant de Lyapunov positif dans des familles à un paramétre d'applications unimodales. C.R. Acad. Sci. Paris Série I 315 (1992), 69-72.
82. M. Tsujii. Positive Lyapunov exponents in families of one-dimensional dynamical systems. Invent. Math. 111 (1993), 113-137.
83. M. Tsujii. On continuity of Bowen-Ruelle-Sinai measures in families of one-dimensional maps. Commun. Math. Phys. 177 (1996), 1-11.
84. J.-C. Yoccoz. Il n'y a pas de contre-exemple de Denjoy analitique. C.R. Acad. Sci. Paris, Série I 298 (1984), 141-144.
85. J.-C. Yoccoz. Personal Communication; see also [Mi2].
86. L.-S. Young. Decay of correlations for certain quadratic maps. Commun. Math. Phys. 146 (1992), 123-138.