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://www.numdam.org/articles/10.1007/s10240-005-0033-2/}
}
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://www.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 1796837 | Zbl 0963.37003

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 2018784 | Zbl 1050.37018

3. A. Avila, C. G. Moreira, Statistical properties of unimodal maps: the quadratic family. Ann. Math., 161 (2005), 827-877. | MR 2153401 | Zbl 1078.37029

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. | MR 2052298 | Zbl 1046.37021

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 799250 | Zbl 0597.58016

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 1132757 | Zbl 0790.58024

8. M. Jacobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys., 81 (1981), 39-88. | MR 630331 | Zbl 0497.58017

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 1182410 | Zbl 0763.58024

10. O. S. Kozlovski, Getting rid of the negative Schwarzian derivative condition. Ann. Math., 152 (2000), 743-762. | MR 1815700 | Zbl 0988.37044

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 1954263

12. A. N. Livsic, The homology of dynamical systems. Usp. Mat. Nauk, 27 (1972), no. 3(165), 203-204. | MR 394768

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 1298717 | Zbl 0821.58014

14. M. Lyubich, Dynamics of quadratic polynomials, I-II. Acta Math., 178 (1997), 185-297. | MR 1459261 | Zbl 0908.58053

15. M. Lyubich, Dynamics of quadratic polynomials, III. Parapuzzle and SBR measure. Astérisque, 261 (2000), 173-200. | MR 1755441 | Zbl 1044.37038

16. M. Lyubich, Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture. Ann. Math., 149 (1999), 319-420. | Zbl 0945.37012

17. M. Lyubich, Almost every real quadratic map is either regular or stochastic. Ann. Math., 156 (2002), 1-78. | MR 1935840

18. R. Mañé, Hyperbolicity, sinks and measures for one-dimensional dynamics. Commun. Math. Phys., 100 (1985), 495-524. | MR 806250 | Zbl 0583.58016

19. M. Martens, W. De Melo, The multipliers of periodic points in one-dimensional dynamics, Nonlinearity, 12 (1999), 217-227. | MR 1677736 | Zbl 0989.37032

20. W. De Melo, S. Van Strien, One-dimensional dynamics. Springer 1993. | MR 1239171 | Zbl 0791.58003

21. J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory. Math. Intell., 19 (1997), 30-32. | Zbl 0883.28004

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 970571 | Zbl 0664.58015

23. T. Nowicki, D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math., 132 (1998), 633-680. | MR 1625708 | Zbl 0908.58016

24. D. Ruelle, A. Wilkinson. Absolutely singular dynamical foliations. Commun. Math. Phys., 219 (2001), 481-487. | MR 1838747 | Zbl 1031.37029

25. M. Shub, D. Sullivan, Expanding endomorphisms of the circle revisited. Ergodic Theory Dyn. Syst., 5 (1985), 285-289. | MR 796755 | Zbl 0583.58022

26. M. Shub, A. Wilkinson, Pathological foliations and removable zero exponents. Invent. Math., 139 (2000), 495-508. | MR 1738057 | Zbl 0976.37013

27. M. Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems. Invent. Math., 111 (1993), 113-137. | MR 1193600 | Zbl 0787.58029