Ergodic attractors

Pugh, Charles; Shub, Michael

doi:10.1090/S0002-9947-1989-0983869-1

Ergodic attractors
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by Charles Pugh and Michael Shub PDF

Trans. Amer. Math. Soc. 312 (1989), 1-54 Request permission

Abstract:

Using the graph transform method, we give a geometric treatment of Pesin’s invariant manifold theory. Beyond deriving the existence, uniqueness, and smoothness results by Fathi, Herman, and Yoccoz our method allows us to do four things: optimally conserve smoothness, deal with endomorphisms, prove absolute continuity of the Pesin laminations, and produce ergodic attractors.

References

D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
N. Bourbaki, Éléments de mathématique. XXV. Première partie. Livre VI: Intégration. Chapitre 6: Intégration vectorielle, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1281, Hermann, Paris, 1959 (French). MR 0124722
Morton Brown, The monotone union of open $n$-cells is an open $n$-cell, Proc. Amer. Math. Soc. 12 (1961), 812–814. MR 126835, DOI 10.1090/S0002-9939-1961-0126835-6
A. Fathi, M.-R. Herman, and J.-C. Yoccoz, A proof of Pesin’s stable manifold theorem, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 177–215. MR 730270, DOI 10.1007/BFb0061417
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822
Ricardo Mañé, Lyapounov exponents and stable manifolds for compact transformations, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 522–577. MR 730286, DOI 10.1007/BFb0061433

Teoria ergodica

Ricardo Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977), 351–370. MR 482849, DOI 10.1090/S0002-9947-1977-0482849-4
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1332–1379, 1440 (Russian). MR 0458490
Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684

Ergodic theory

Charles Pugh and Michael Shub, Ergodicity of Anosov actions, Invent. Math. 15 (1972), 1–23. MR 295390, DOI 10.1007/BF01418639

Differentiability and continuity of invariant manifolds

Charles C. Pugh, The $C^{1+\alpha }$ hypothesis in Pesin theory, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 143–161. MR 743817
M. S. Raghunathan, A proof of Oseledec’s multiplicative ergodic theorem, Israel J. Math. 32 (1979), no. 4, 356–362. MR 571089, DOI 10.1007/BF02760464
David Ruelle and Michael Shub, Stable manifolds for maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 389–392. MR 591198
David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243–290. MR 647807, DOI 10.2307/1971392
Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255, DOI 10.1007/978-1-4757-1947-5
Anatole Katok, Jean-Marie Strelcyn, F. Ledrappier, and F. Przytycki, Invariant manifolds, entropy and billiards; smooth maps with singularities, Lecture Notes in Mathematics, vol. 1222, Springer-Verlag, Berlin, 1986. MR 872698, DOI 10.1007/BFb0099031