Random perturbations and statistical properties of Hénon-like maps

Michael Benedicks; Marcelo Viana

Annales de l'I.H.P. Analyse non linéaire (2006)

  • Volume: 23, Issue: 5, page 713-752
  • ISSN: 0294-1449

How to cite

top

Benedicks, Michael, and Viana, Marcelo. "Random perturbations and statistical properties of Hénon-like maps." Annales de l'I.H.P. Analyse non linéaire 23.5 (2006): 713-752. <http://eudml.org/doc/78709>.

@article{Benedicks2006,
author = {Benedicks, Michael, Viana, Marcelo},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {dynamics; non-uniform hyperbolic; strange attractor; random perturbation; Hénon map},
language = {eng},
number = {5},
pages = {713-752},
publisher = {Elsevier},
title = {Random perturbations and statistical properties of Hénon-like maps},
url = {http://eudml.org/doc/78709},
volume = {23},
year = {2006},
}

TY - JOUR
AU - Benedicks, Michael
AU - Viana, Marcelo
TI - Random perturbations and statistical properties of Hénon-like maps
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2006
PB - Elsevier
VL - 23
IS - 5
SP - 713
EP - 752
LA - eng
KW - dynamics; non-uniform hyperbolic; strange attractor; random perturbation; Hénon map
UR - http://eudml.org/doc/78709
ER -

References

top
  1. [1] J.F. Alves, V. Araújo, Stochastic stability for robust classes of non-uniformly expanding maps, Astérisque. 
  2. [2] Andronov A., Pontryagin L., Systèmes grossiers, Dokl. Akad. Nauk USSR14 (1937) 247-251. Zbl0016.11301
  3. [3] Araújo V., Attractors and time averages for random maps, Ann. Inst. H. Poincaré Anal. Non Linéaire17 (2000) 307-369. Zbl0974.37036MR1771137
  4. [4] Arnold L., Random Dynamical Systems, Springer-Verlag, 1998. Zbl0834.58026MR1723992
  5. [5] A. Avila, C.G. Moreira, Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative, Astérisque. Zbl1046.37021MR2052298
  6. [6] Baladi V., Viana M., Strong stochastic stability and rate of mixing for unimodal maps, Ann. Sci. École Norm. Sup.29 (1996) 483-517. Zbl0868.58051MR1386223
  7. [7] Benedicks M., Carleson L., The dynamics of the Hénon map, Ann. of Math.133 (1991) 73-169. Zbl0724.58042MR1087346
  8. [8] Benedicks M., Viana M., Solution of the basin problem for Hénon-like attractors, Invent. Math.143 (2001) 375-434. Zbl0967.37023MR1835392
  9. [9] Benedicks M., Young L.-S., Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps, Ergodic Theory Dynam. Systems12 (1992) 13-37. Zbl0769.58051MR1162396
  10. [10] Benedicks M., Young L.-S., SBR-measures for certain Hénon maps, Invent. Math.112 (1993) 541-576. Zbl0796.58025MR1218323
  11. [11] Benedicks M., Young L.-S., Markov extensions and decay of correlations for certain Hénon maps, Astérisque261 (2000) 13-56. Zbl1044.37013MR1755436
  12. [12] P. Collet, Ergodic properties of some unimodal mappings of the interval, Technical report, Institute Mittag-Leffler, 1984. 
  13. [13] de Melo W., van Strien S., One-Dimensional Dynamics, Springer-Verlag, 1993. Zbl0791.58003MR1239171
  14. [14] Díaz L.J., Rocha J., Viana M., Strange attractors in saddle-node cycles: prevalence and globality, Invent. Math.125 (1996) 37-74. Zbl0865.58034MR1389960
  15. [15] Hayashi S., Connecting invariant manifolds and the solution of the stability andΩ-stability conjectures for flows, Ann. of Math.145 (1997) 81-137. Zbl0871.58067
  16. [16] Katok A., Kifer Yu., Random perturbations of transformations of an interval, J. Anal. Math.47 (1986) 193-237. Zbl0616.60064MR874051
  17. [17] Keller G., Stochastic stability in some chaotic dynamical systems, Monatsh. Math.94 (1982) 313-333. Zbl0496.58010MR685377
  18. [18] Kifer Yu., Ergodic Theory of Random Perturbations, Birkhäuser, 1986. MR884892
  19. [19] Kifer Yu., Random Perturbations of Dynamical Systems, Birkhäuser, 1988. Zbl0659.58003MR1015933
  20. [20] Mañé R., A proof of the stability conjecture, Publ. Math. I.H.E.S.66 (1988) 161-210. Zbl0678.58022MR932138
  21. [21] Metzger R., Stochastic stability for contracting Lorenz maps, Comm. Math. Phys.212 (2000) 277-296. Zbl1052.37018MR1772247
  22. [22] Mora L., Viana M., Abundance of strange attractors, Acta Math.171 (1993) 1-71. Zbl0815.58016MR1237897
  23. [23] Palis J., Smale S., Structural stability theorems, in: Global Analysis, Berkeley, 1968, Proc. Sympos. Pure Math., vol. XIV, Amer. Math. Soc., 1970, pp. 223-232. Zbl0214.50702MR267603
  24. [24] Palis J., Takens F., Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993. Zbl0790.58014MR1237641
  25. [25] Pesin Ya., Families of invariant manifolds corresponding to non-zero characteristic exponents, Math. USSR Izv.10 (1976) 1261-1302. Zbl0383.58012
  26. [26] Pugh C., Shub M., Ergodic attractors, Trans. Amer. Math. Soc.312 (1989) 1-54. Zbl0684.58008MR983869
  27. [27] Robbin J., A structural stability theorem, Ann. of Math.94 (1971) 447-493. Zbl0224.58005MR287580
  28. [28] Robinson C., Structural stability of vector fields, Ann. of Math.99 (1974) 154-175, Errata, Ann. of Math.101 (1975) 368. Zbl0275.58012MR334283
  29. [29] Rokhlin V.A., On the fundamental ideas of measure theory, Amer. Math. Soc. Transl.10 (1962) 1-52, Transl. from, Mat. Sb.25 (1949) 107-150. Zbl0174.45501MR30584
  30. [30] Rudin W., Real and Complex Analysis, McGraw-Hill, 1987. Zbl0925.00005MR924157
  31. [31] Sinai Ya., Gibbs measures in ergodic theory, Russian Math. Surveys27 (1972) 21-69. Zbl0255.28016MR399421
  32. [32] Thieullen Ph., Tresser C., Young L.-S., Positive Lyapunov exponent for generic one-parameter families of unimodal maps, J. Anal. Math.64 (1994) 121-172. Zbl0821.58015MR1303510
  33. [33] Wang Q., Young L.-S., Strange attractors with one direction of instability, Comm. Math. Phys.218 (2001) 1-97. Zbl0996.37040MR1824198
  34. [34] Young L.-S., Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems6 (1986) 311-319. Zbl0633.58023MR857204

NotesEmbed ?

top

You must be logged in to post comments.