Statistical properties of unimodal maps
Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 1-67.

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.

@article{PMIHES_2005__101__1_0,
     author = {Avila, Artur and Moreira, Carlos Gustavo},
     title = {Statistical properties of unimodal maps},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--67},
     publisher = {Springer},
     volume = {101},
     year = {2005},
     doi = {10.1007/s10240-005-0033-2},
     zbl = {1078.37030},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-005-0033-2/}
}
TY  - JOUR
AU  - Avila, Artur
AU  - Moreira, Carlos Gustavo
TI  - Statistical properties of unimodal maps
JO  - Publications Mathématiques de l'IHÉS
PY  - 2005
SP  - 1
EP  - 67
VL  - 101
PB  - Springer
UR  - http://archive.numdam.org/articles/10.1007/s10240-005-0033-2/
DO  - 10.1007/s10240-005-0033-2
LA  - en
ID  - PMIHES_2005__101__1_0
ER  - 
%0 Journal Article
%A Avila, Artur
%A Moreira, Carlos Gustavo
%T Statistical properties of unimodal maps
%J Publications Mathématiques de l'IHÉS
%D 2005
%P 1-67
%V 101
%I Springer
%U http://archive.numdam.org/articles/10.1007/s10240-005-0033-2/
%R 10.1007/s10240-005-0033-2
%G en
%F PMIHES_2005__101__1_0
Avila, Artur; Moreira, Carlos Gustavo. Statistical properties of unimodal maps. Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 1-67. doi : 10.1007/s10240-005-0033-2. http://archive.numdam.org/articles/10.1007/s10240-005-0033-2/

1. V. Arnold, Dynamical systems, in Development of mathematics 1950-2000, pp. 33-61, Birkhäuser, Basel 2000. | MR | Zbl

2. A. Avila, M. Lyubich, W. De Melo, Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math., 154 (2003), 451-550. | MR | Zbl

3. A. Avila, C. G. Moreira, Statistical properties of unimodal maps: the quadratic family. Ann. Math., 161 (2005), 827-877. | MR | Zbl

4. A. Avila, C. G. Moreira, Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative. Geometric methods in dynamics. I. Astérisque, 286 (2003), 81-118. | Numdam | MR | Zbl

5. A. Avila, C. G. Moreira, Phase-Parameter relation and sharp statistical properties for general families of unimodal maps, preprint (http://www.arXiv.org), to appear in Contemp. Math., volume on “Geometry and Dynamics”, ed. by E. Ghys, J. Eells, M. Lyubich, J. Palis, J. Seade.

6. M. Benedicks, L. Carleson, On iterations of 1-ax 2 on (-1,1). Ann. Math., 122 (1985), 1-25. | MR | Zbl

7. A. M. Blokh, M. Yu. Lyubich, Measurable dynamics of S-unimodal maps of the interval. Ann. Sci. Éc. Norm. Supér., IV. Sér., 24 (1991), 545-573. | Numdam | MR | Zbl

8. M. Jacobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys., 81 (1981), 39-88. | MR | Zbl

9. G. Keller, T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Commun. Math. Phys., 149 (1992), 31-69. | MR | Zbl

10. O. S. Kozlovski, Getting rid of the negative Schwarzian derivative condition. Ann. Math., 152 (2000), 743-762. | MR | Zbl

11. O. S. Kozlovski, Axiom A maps are dense in the space of unimodal maps in the Ck topology. Ann. Math., 157 (2003), 1-43. | MR

12. A. N. Livsic, The homology of dynamical systems. Usp. Mat. Nauk, 27 (1972), no. 3(165), 203-204. | MR

13. M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math., 140 (1994), 347-404. Note on the geometry of generalized parabolic towers. Manuscript (2000) (http://www.arXiv.org). | MR | Zbl

14. M. Lyubich, Dynamics of quadratic polynomials, I-II. Acta Math., 178 (1997), 185-297. | MR | Zbl

15. M. Lyubich, Dynamics of quadratic polynomials, III. Parapuzzle and SBR measure. Astérisque, 261 (2000), 173-200. | MR | Zbl

16. M. Lyubich, Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture. Ann. Math., 149 (1999), 319-420. | Zbl

17. M. Lyubich, Almost every real quadratic map is either regular or stochastic. Ann. Math., 156 (2002), 1-78. | MR

18. R. Mañé, Hyperbolicity, sinks and measures for one-dimensional dynamics. Commun. Math. Phys., 100 (1985), 495-524. | MR | Zbl

19. M. Martens, W. De Melo, The multipliers of periodic points in one-dimensional dynamics, Nonlinearity, 12 (1999), 217-227. | MR | Zbl

20. W. De Melo, S. Van Strien, One-dimensional dynamics. Springer 1993. | MR | Zbl

21. J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory. Math. Intell., 19 (1997), 30-32. | Zbl

22. J. Milnor, W. Thurston, On iterated maps of the interval, Dynamical Systems, Proc. U. Md., 1986-87, ed. by J. Alexander. Lect. Notes Math., 1342 (1988), 465-563. | MR | Zbl

23. T. Nowicki, D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math., 132 (1998), 633-680. | MR | Zbl

24. D. Ruelle, A. Wilkinson. Absolutely singular dynamical foliations. Commun. Math. Phys., 219 (2001), 481-487. | MR | Zbl

25. M. Shub, D. Sullivan, Expanding endomorphisms of the circle revisited. Ergodic Theory Dyn. Syst., 5 (1985), 285-289. | MR | Zbl

26. M. Shub, A. Wilkinson, Pathological foliations and removable zero exponents. Invent. Math., 139 (2000), 495-508. | MR | Zbl

27. M. Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems. Invent. Math., 111 (1993), 113-137. | MR | Zbl

Cité par Sources :