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Katok, A.

Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. (English) Zbl 0445.58015

Publ. Math., Inst. Hautes Étud. Sci. 51, 137-173 (1980).
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scan of Zbl 0445.58015

Cited in 21 Reviews
Cited in 497 Documents

MSC:

37A99 Ergodic theory
37C75 Stability theory for smooth dynamical systems
37D99 Dynamical systems with hyperbolic behavior
37G99 Local and nonlocal bifurcation theory for dynamical systems
28D20 Entropy and other invariants

Keywords:

epsilon trajectories; invariant measures; periodic points; entropy; pseudo-orbit tracing; Lyapunov exponents; hyperbolic period points; transversal homoclinic point
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Full Text: DOI Numdam EuDML

References:

[1] R. Bowen, Topological entropy and Axiom A,Proc. Symp. Pure Math., A.M.S., Providence R.I.,14 (1970), 23–41. · Zbl 0207.54402
[2] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms,Trans. Amer. Math. Soc.,154 (1971). 377–397. · Zbl 0212.29103
[3] R. Bowen, Periodic orbits for hyperbolic flows,Amer. J. Math.,94 (1972), 1–30. · Zbl 0254.58005 · doi:10.2307/2373590
[4] D. V. Anosov, On certain class of invariant sets of smooth dynamical systems (in Russian),Proc. 5th International Conf. on Non-linear Oscillations, vol. 2, Kiev, 1970, 39–45. · Zbl 0243.34085
[5] A. B. Katok, Dynamical systems with hyperbolic structure (in Russian),Ninth Summer Math. School, Kier, published by Math. Inst. of the Ukrainian Acad. of Sci., 1972; revised edition Kiev, Naukova Dumka, 1976, 125–211; to be translated into English.
[6] A. B. Katok, Local properties of hyperbolic sets (in Russian). Addition to the Russian translation ofZ. Nitecki,Differentiable Dynamics, Moscow, Mir, 1975, 214–232.
[7] R. Bowen, Some systems with unique equilibrium state,Math. Systems Theory,8 (1975), 193–203. · Zbl 0299.54031 · doi:10.1007/BF01762666
[8] Ja. B. Pesin, Families of invariant manifolds corresponding to non-zero characteristic exponents,Math. of the USSR-Izvestija,10 (1976), 6, 1261–1305; translated from Russian. · Zbl 0383.58012 · doi:10.1070/IM1976v010n06ABEH001835
[9] Ja. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,Russian Math. Surveys,32 (1977), 4, 55–114; translated from Russian. · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639
[10] Ja. B. Pesin, Description of {\(\pi\)}-partition of a diffeomorphism with invariant measure,Math. Notes of the USSR Acad. of Sci.,22 (1976), 1, 506–514; translated from Russian. · Zbl 0423.58013 · doi:10.1007/BF01147690
[11] Ja. B. Pesin, Geodesic flows on closed Riemannian surfaces without focal points,Math. of the USSR-Izvestijs,11 (1977), 6, 1195–1228; translated from Russian. · Zbl 0399.58010 · doi:10.1070/IM1977v011n06ABEH001766
[12] M. I. Brin, Ja. B. Pesin, Partially hyperbolic dynamical systems,Math. of the USSR-Izvestija,8 (1974), 1, 177–218; translated from Russian. · Zbl 0309.58017 · doi:10.1070/IM1974v008n01ABEH002101
[13] V. I. Oseledec, Multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,Trans. Moscow Math. Soc.,19 (1968), 197–221; translated from Russian.
[14] M. Raghunathan, A proof of Oseledec’s multiplicative ergodic theorem,Israel J. Math., to appear. · Zbl 0415.28013
[15] M. I. Zaharevitch, Characteristic exponents and vector ergodic theorem (in Russian),Leningrad Univ. Vestnik, Math. Mech. Astr, 7–2, 1978, 28–34.
[16] S. Katok, The estimation from above for the topological entropy of a diffeomorphism,Proc. Conf. on Dynamical Syst., Evanston, 1979, to appear inLecture Notes in Math.
[17] D. Ruelle, An inequality for the entropy of differentiable maps,Bol. Soc. Bras. Mat.,9 (1978), 83–87. · Zbl 0432.58013 · doi:10.1007/BF02584795
[18] R. Bowen, Entropy for group automorphisms and homogenous spaces,Trans. Amer. Math. Soc.,153 (1971), 401–414. · Zbl 0212.29201 · doi:10.1090/S0002-9947-1971-0274707-X
[19] E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems,Math. of the USSR-Izvestija,5 (1971), 2, 337–378; translated from Russian. · Zbl 0248.58007 · doi:10.1070/IM1971v005n02ABEH001050
[20] S. Smale, Diffeomorphisms with many periodic points,Diff. and Comb. Topology, Princeton Univ. Press, Princeton, 1965, 63–80.
[21] D. Ruelle, Ergodic theory of differentiable dynamical systems,Publ. Math. I.H.E.S.,50 (1979), 27–58. · Zbl 0426.58014
[22] A. Katok,Smooth Ergodic Theory, Lecture Notes, University of Maryland, in preparation.
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