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

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
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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
[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
[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
[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
[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
[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
[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
[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
[18] R. Bowen, Entropy for group automorphisms and homogenous spaces,Trans. Amer. Math. Soc.,153 (1971), 401–414. · Zbl 0212.29201
[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
[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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