×

zbMATH — the first resource for mathematics

Multidimensional nonhyperbolic attractors. (English) Zbl 1037.37016
Summary: We construct smooth transformations and diffeomorphisms exhibiting nonuniformly hyperbolic attractors with multidimensional sensitiveness on initial conditions: typical orbits in the basin of attraction have several expanding directions. These systems also illustrate a new robust mechanism of sensitive dynamics: despite the nonuniform character of the expansion, the attractor persists in a full neighbourhood of the initial map.

MSC:
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] M. Benedicks, L. Carleson, On iterations of 1 x 2 on (, 1),Ann. Math. 122 (1985), 1–25. · Zbl 0597.58016
[2] M. Benedicks, L. Carleson, The dynamics of the Hénon map,Ann. Math. 133 (1991), 73–169. · Zbl 0724.58042
[3] M. Benedicks, L.-S. Young, SBR-measures for certain Hénon maps,Invent. Math. 112–3 (1993), 541–576. · Zbl 0796.58025
[4] C. Bonatti, L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms,Ann. Math. 143 (1996), 357–396. · Zbl 0852.58066
[5] P. Collet, J.-P. Eckmann, On the abundance of aperiodic behaviour,Comm. Math. Phys. 73 (1980), 115–160. · Zbl 0441.58011
[6] M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds,Lect. Notes Math. 583 (1977), Springer Verlag. · Zbl 0355.58009
[7] M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps,Comm. Math. Phys. 81 (1981), 39–88. · Zbl 0497.58017
[8] R. Mañé, Contributions to the stability conjecture,Topology 17 (1978), 383–396. · Zbl 0405.58035
[9] W. de Melo, S. van Strien, One-Dimensional Dynamics, Springer Verlag, 1993. · Zbl 0791.58003
[10] L. Mora, M. Viana, Abundance of strange attractors,Acta Math. 171 (1993), 1–71. · Zbl 0815.58016
[11] T. Nowicki, A positive Lyapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity,Ergod. Th. & Dynam. Sys. 8 (1988), 425–435. · Zbl 0638.58021
[12] M. Shub, Topologically transitive diffeomorphisms on T4,Lect. Notes in Math. 206 (1971), 39, Springer Verlag.
[13] D. Singer, Stable orbits and bifurcations of maps of the interval,SIAM J. Appl. Math. 35 (1978), 260–267. · Zbl 0391.58014
[14] S. Smale, Differentiable dynamical systems,Bull. Am. Math. Soc. 73 (1967), 747–817. · Zbl 0202.55202
[15] M. Viana, Strange attractors in higher dimensions,Bull. Braz. Math. Soc. 24 (1993), 13–62. · Zbl 0784.58044
[16] L.-S. Young, Some open sets of nonuniformly hyperbolic cocycles,Ergod. Th. & Dynam. Sys. 13 (1993), 409–415. · Zbl 0797.58041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.