Statistical stability for Hénon maps of the Benedicks–Carleson type
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 595-637.

We consider the family of Hénon maps in the plane and show that the SRB measures vary continuously in the weak∗ topology within the set of Benedicks–Carleson parameters.

DOI : 10.1016/j.anihpc.2009.09.009
Classification : 37C40, 37C75, 37D25
Mots-clés : Hénon attractor, SRB measure, Statistical stability 
@article{AIHPC_2010__27_2_595_0,
     author = {Alves, Jos\'e F. and Carvalho, Maria and Freitas, Jorge Milhazes},
     title = {Statistical stability for {H\'enon} maps of the {Benedicks{\textendash}Carleson} type},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {595--637},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.09.009},
     mrnumber = {2595193},
     zbl = {1205.37040},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.009/}
}
TY  - JOUR
AU  - Alves, José F.
AU  - Carvalho, Maria
AU  - Freitas, Jorge Milhazes
TI  - Statistical stability for Hénon maps of the Benedicks–Carleson type
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 595
EP  - 637
VL  - 27
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.009/
DO  - 10.1016/j.anihpc.2009.09.009
LA  - en
ID  - AIHPC_2010__27_2_595_0
ER  - 
%0 Journal Article
%A Alves, José F.
%A Carvalho, Maria
%A Freitas, Jorge Milhazes
%T Statistical stability for Hénon maps of the Benedicks–Carleson type
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 595-637
%V 27
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.009/
%R 10.1016/j.anihpc.2009.09.009
%G en
%F AIHPC_2010__27_2_595_0
Alves, José F.; Carvalho, Maria; Freitas, Jorge Milhazes. Statistical stability for Hénon maps of the Benedicks–Carleson type. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 595-637. doi : 10.1016/j.anihpc.2009.09.009. http://www.numdam.org/articles/10.1016/j.anihpc.2009.09.009/

[1] J.F. Alves, M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems 22 (2002), 1-32 | MR | Zbl

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

[3] M. Benedicks, L. Carleson, The dynamics of the Hénon map, Ann. of Math. 133 (1991), 73-169 | MR | Zbl

[4] M. Benedicks, M. Viana, Random perturbations and statistical properties of Hénon-like maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 no. 5 (2006), 713-752 | EuDML | Numdam | MR | Zbl

[5] M. Benedicks, M. Viana, Solution of the basin problem for Hénon-like attractors, Invent. Math. 143 (2001), 375-434 | MR | Zbl

[6] M. Benedicks, L. Young, Sinai–Bowen–Ruelle measures for certain Hénon maps, Invent. Math. 112 (1993), 541-576 | EuDML | MR | Zbl

[7] M. Benedicks, L. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque 261 (2000), 13-56 | Numdam | MR | Zbl

[8] C. Bonatti, L.J. Díaz, M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer-Verlag (2005) | MR | Zbl

[9] R. Bowen, Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. vol. 470, Springer-Verlag (1975) | MR | Zbl

[10] P. Collet, J. Eckmann, On the abundance of aperiodic behavior for maps on the interval, Comm. Math. Phys. 73 (1980), 115-160 | MR | Zbl

[11] P. Collet, J.P. Eckmann, On the abundance of aperiodic behavior for maps on the interval, Bull. Amer. Math. Soc. 3 no. 1 (1980), 699-700 | MR | Zbl

[12] P. Collet, J.P. Eckmann, Positive Lyapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems 3 (1983), 13-46 | MR | Zbl

[13] J.M. Freitas, Continuity of SRB measure and entropy for Benedicks–Carleson quadratic maps, Nonlinearity 18 (2005), 831-854 | MR | Zbl

[14] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), 69-77 | MR | Zbl

[15] M. Hirsch, S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press Inc. (1974) | MR | Zbl

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

[17] S. Luzzatto, M. Viana, Parameter exclusions in Hénon-like systems, Russian Math. Surveys 58 (2003), 1053-1092 | Zbl

[18] L. Mora, M. Viana, Abundance of strange attractors, Acta Math. 171 (1993), 1-71 | MR | Zbl

[19] S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9-18 | MR | Zbl

[20] S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101-152 | EuDML | Numdam | MR | Zbl

[21] J.B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR Izv. 10 (1978), 1261-1305 | Zbl

[22] D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 27-58 | EuDML | Numdam | MR | Zbl

[23] M. Rychlik, E. Sorets, Regularity and other properties of absolutely continuous invariant measures for the quadratic family, Comm. Math. Phys. 150 no. 2 (1992), 217-236 | MR | Zbl

[24] H. Thunberg, Unfolding of chaotic unimodal maps and the parameter dependence of natural measures, Nonlinearity 14 no. 2 (2001), 323-337 | MR | Zbl

[25] M. Tsujii, On continuity of Bowen–Ruelle–Sinai measures in families of one dimensional maps, Comm. Math. Phys. 177 no. 1 (1996), 1-11 | MR | Zbl

[26] R. Ures, On the approximation of Hénon-like attractors by homoclinic tangencies, Ergodic Theory Dynam. Systems 15 (1995), 1223-1229 | MR | Zbl

[27] R. Ures, Hénon attractors: SRB measures and Dirac measures for sinks, 1st International Conference on Dynamical Systems – A Tribute to Ricardo Mañé, Montevideo, Uruguay, 1995, Pitman Res. Notes Math. Ser. vol. 362, Longman, Harlow (1996), 214-219 | Zbl

[28] Q. Wang, L.S. Young, Strange attractors with one direction of instability, Comm. Math. Phys. 218 (2001), 1-97 | MR | Zbl

[29] L.S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147 (1998), 585-650 | MR | Zbl

Cité par Sources :