Iterations of holomorphic collet-eckmann maps: Conformai, and invariant measures. Appendix: on non-renormalizable quadratic polynomials

Transactions of the American Mathematical Society

Volumn 350, Issue 2, 1998, Pages 717-742

  Przytycki, Feliks ^a  

^a INSTITUTE OF MATHEMATICS   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646869539     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-98-01890-x     Document Type: Article

References (30)

1. BL] M. Bloch and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. Éc. Norm. Sup. (4) 24 (1991), 545-573.
2. CE] P. Collet and J.-P. Eckmann, Positive Lyapunov exponents and absolute continuity for maps of the interval, Ergodic Th. and Dyn. Sys. 3 (1983), 13-46. MR 85j:58092
3. CEbook] _Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Basel-Boston-Stuttgart, 1980. MR 82j:58078
4. DPU] M. Denker, F. Przytycki, and M. Urbanski, On the transfer operator for rational functions on the Riemann sphere, Ergodic Th. and Dyn. Sys. 16 (1996), 255-266. MR 97e:58197
5. DU1] M. Denker and M. Urbanski, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365-384. MR 92f:50097
6. DU2] _ Capacity of parabolic Julia sets, Math. Zeit. 211 (1992), 73-86. MR 93j:30022
7. n, Lecture Notes in Math. vol. 481. Springer-Verlag, Berlin 1975. MR 56:15866
8. Guckenheimer] J. Guckenheimer, Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), 133-160. MR 82c:58037
9. GJ] J. Guckenheimer and S. Johnson, Distortion of S-unimodal maps, Ann. of Math. 132 (1990), 71-130. MR 91g:58157
10. GS1] J. Graczyk and G. Świa̧tek, Induced expansion for quadratic polynomials, Ann. Sci. Éc. Norm. Sup. (4) 29 (1996), 399-482. CMP 96:11
11. GS2] J. Graczyk and G. Świa̧tek, Holomorphic box mappings, Preprint IHES/M/1996/76.
12. H] E. Hille, Analytic Function Theory, Ginn and Company, Boston 1962. MR 34:1490
13. K] A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. Math. IHES 51 (1980), 137-173. MR 81i:28022
14. L] M. Lyubich, Geometry of quadratic polynomials: moduli, rigidity and local connectivity, Preprint SUNY at Stony Brook, IMS 1993/9.
15. LMi] M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), 425-457. MR 93h:58080
16. M] R. Mañé, On a theorem of Fatou, Bol. Soc. Bras. Mat. 24 (1993), 1-12. MR 94g:58188
17. N] T. Nowicki, Some dynamical properties of S-unimodal maps, Fund. Math. 142 (1993), 45-57. MR 94c:58111
18. McM] C. McMullen, Complex Dynamics and Renormalization, Ann. of Math. Studies 135, Princeton University Press, 1994. MR 96b:58097
19. NS] T. Nowicki and S. van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math. 105 (1991), 123-136. MR 93b:58094
20. Prado] E. Prado, Ergodicity of conformal measures for quadratic polynomials, Manuscript, May 23, 1994.
21. P1] F. Przytycki, Lyapunov characteristic exponents are non-negative, Proc. Amer. Math. Soc. 119(1) (1993), 309-317. MR 93k:58193
22. P2] _Invariant measures for iterations of holomorphic maps, In "Linear and Complex Analysis. Problem Book 3" Part II, Eds. V. P. Havin, N. K. Nikolski. Lect. Notes in Math. 1574, Springer (1994), 450-454. MR 96c:00001b
23. P3] _On measure and Hausdorff dimension of Julia sets for holomorphic Collet-Eckmann maps, In "International conference on dynamical systems, Montevideo 1995 - a tribute to Ricardo Mane", (Eds) F. Ledrappier, J. Lewowicz, S. Newhouse, Pitman Res. Notes in Math. 362, Longman (1996), 167-181.
24. PUbook] F. Przytycki and M. Urbariski, To appear.
25. PUZ] F. Przytycki, M. Urbariski, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I. Ann. of Math. 130 (1989), 1-40. MR 91i:58115
26. R] M. Rees, Positive measure sets of ergodic rational maps, Ann. Sci. Éc. Norm. Sup. (4) 19 (1986), 383-407. MR 88g:58100
27. S] D. Sullivan, Conformal dynamical systems, In "Geometric Dynamics", Lec. Notes in Math., vol. 1007, Springer, New York (1983), 725-752. MR 85m:58112
28. Shi] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia set, Preprint SUNY at Stony Brook, IMS 1991/7.
29. U] M. Urbański, Rational functions with no recurrent critical points, Ergodic Th. and Dyn. Sys. 14.2 (1994), 391-414. MR 95g:58191
30. Y] J.-Ch. Yoccoz, Talks on several conferences.