Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative

Avila, Artur ; Moreira, Carlos Gustavo

Geometric methods in dynamics (I) : Volume in honor of Jacob Palis, Astérisque, no. 286 (2003), pp. 81-118.

@incollection{AST_2003__286__81_0,
     author = {Avila, Artur and Moreira, Carlos Gustavo},
     title = {Statistical properties of unimodal maps: smooth families with negative {Schwarzian} derivative},
     booktitle = {Geometric methods in dynamics (I) : Volume in honor of Jacob Palis},
     editor = {de Melo, Wellington and Viana, Marcelo and Yoccoz, Jean-Christophe},
     series = {Ast\'erisque},
     pages = {81--118},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {286},
     year = {2003},
     mrnumber = {2052298},
     zbl = {1046.37021},
     language = {en},
     url = {http://archive.numdam.org/item/AST_2003__286__81_0/}
}

TY  - CHAP
AU  - Avila, Artur
AU  - Moreira, Carlos Gustavo
TI  - Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative
BT  - Geometric methods in dynamics (I) : Volume in honor of Jacob Palis
AU  - Collectif
ED  - de Melo, Wellington
ED  - Viana, Marcelo
ED  - Yoccoz, Jean-Christophe
T3  - Astérisque
PY  - 2003
SP  - 81
EP  - 118
IS  - 286
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_2003__286__81_0/
LA  - en
ID  - AST_2003__286__81_0
ER  -

%0 Book Section
%A Avila, Artur
%A Moreira, Carlos Gustavo
%T Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative
%B Geometric methods in dynamics (I) : Volume in honor of Jacob Palis
%A Collectif
%E de Melo, Wellington
%E Viana, Marcelo
%E Yoccoz, Jean-Christophe
%S Astérisque
%D 2003
%P 81-118
%N 286
%I Société mathématique de France
%U http://archive.numdam.org/item/AST_2003__286__81_0/
%G en
%F AST_2003__286__81_0

Avila, Artur; Moreira, Carlos Gustavo. Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative, dans Geometric methods in dynamics (I) : Volume in honor of Jacob Palis, Astérisque, no. 286 (2003), pp. 81-118. http://archive.numdam.org/item/AST_2003__286__81_0/

Bibliographie
Cité par

[Ar] V. Arnold. Dynamical systems. In "Development of mathematics 1950-2000", 33-61, Birkhäuser, Basel, 2000. | DOI | MR | Zbl

[A] A. Avila. Bifurcations of unimodal maps: the topologic and metric picture. IMPA Thesis (2001) www.math.sunysb.edu/~artur/. | Zbl

[ALM] A. Avila, M. Lyubich and W. De Melo. Regular or stochastic dynamics in real analytic families of unimodal maps. Preprint IMS at Stony Brook, #2001/15. To appear in Invent. Math. | MR | Zbl

[AM1] A. Avila, C. G. Moreira. Statistical properties of unimodal maps: the quadratic family. Preprint www.arXiv.org. To appear in Annals of Math. | MR | Zbl

[AM2] A. Avila, C. G. Moreira. Statistical properties of unimodal maps: physical measures, periodic orbits and pathological laminations. Preprint www.arXiv.org. | DOI | MR | Zbl

[AM3] A. Avila, C. G. Moreira. Quasisymmetric robustness of the Collet-Eckmann condition in the quadratic family. Preprint www.arXiv.org. | DOI | MR | Zbl

[AM4] A. Avila, C. G. Moreira. Phase-Parameter relation and sharp statistical properties in general families of unimodal maps. Preprint www.arXiv.org. | DOI | MR | Zbl

[BBM] V. Baladi, M. Benedicks and V. Maume. Almost sure rates of mixing for i.i.d. unimodal maps. Ann. Sci. Ecole Norm. Sup. (4), v. 35 (2002), no. 1, 77-126. | DOI | EuDML | Numdam | MR | Zbl

[BV] V. Baladi and M. Viana. Strong stochastic stability and rate of mixing for unimodal maps. Ann. Sci. Ecole Norm. Sup. (4), v. 29 (1996), no. 4, 483-517. | DOI | EuDML | Numdam | MR | Zbl

[BC] M. Benedicks and L. Carleson. On iterations of $1 - a x^{2}$ on $(- 1, 1)$ . Ann. Math., v. 122 (1985), 1-25. | MR | Zbl

[GS] J. Graczyk and G. Swiatek. Generic hyperbolicity in the logistic family. Ann. of Math., v. 146 (1997), 1-52. | DOI | MR | Zbl

[J] M. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys., v. 81 (1981), 39-88. | DOI | MR | Zbl

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

[K] O. S. Kozlovski. Structural stability in one-dimensional dynamics. Thesis (1998).

[L1] M. Lyubich. Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math, 140 (1994), 347-404. | DOI | MR | Zbl

[L2] M. Lyubich. Dynamics of quadratic polynomials, I-II. Acta Math., 178 (1997), 185-297. | DOI | MR | Zbl

[L3] M. Lyubich. Dynamics of quadratic polynomials, III. Parapuzzle and SBR measure. Preprint IMS at Stony Brook, # 1995/5. Astérisque, v. 261 (2000), 173-200. | Numdam | MR | Zbl

[L4] M. Lyubich. Almost every real quadratic map is either regular or stochastic. Ann. of Math. (2) 156 (2002), no. 1, 1-78. | DOI | MR | Zbl

[MN] M. Martens and T. Nowicki. Invariant measures for Lebesgue typical quadratic maps. Preprint IMS at Stony Brook, # 1996/6. Astérisque, v. 261 (2000), 239-252. | Numdam | MR | Zbl

[MvS] W. De Melo and S. Van Strien. One-dimensional dynamics. Springer, 1993. | MR | Zbl

[NP1] T. Nowicki and F. Przytycki. The conjugacy of Collet-Eckmann's map of the interval with the tent map is Hölder continuous. Ergodic Theory Dynam. Systems 9 (1989), no. 2, 379-388. | DOI | MR | Zbl

[NP2] T. Nowicki and F. Przytycki. Topological invariance of the Collet-Eckmann property for S-unimodal maps. Fund. Math. 155 (1998), no. 1, 33-43. | EuDML | MR | Zbl

[NS] T. Nowicki and D. Sands. Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132 (1998), no. 3, 633-680. | DOI | MR | Zbl

[Pa] J. Palis. A global view of dynamics and a conjecture of the denseness of finitude of attractors. Astérisque, v. 261 (2000), 335-347. | Numdam | MR | Zbl

[T1] M. Tsujii. Positive Lyapunov exponents in families of one dimensional dynamical systems. Invent. Math. 111 (1993), 113-137. | DOI | EuDML | MR | Zbl

[T2] M. Tsujii. Small random perturbations of one dimensional dynamical systems and Margulis-Pesin entropy formula. Random & Comput. Dynamics. Vol.1 No.1 59-89, (1992). | MR | Zbl

[Y] L.-S. Young. Decay of correlations for certain quadratic maps. Comm. Math. Phys., 146 (1992), 123-138. | DOI | MR | Zbl