×

Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities. (English) Zbl 0756.58038

For uniformly hyperbolic systems Anosov has shown a strong shadowing property. For non-uniformly hyperbolic systems with singularities we introduce the weaker notion of essential \(\varepsilon_ i\) pseudo-orbits and prove that each essential \(\varepsilon_ i\) pseudo-orbit is \(\delta_ i\) shadowed by a unique point. We use this fact to construct Markov partitions. Our construction is inspired by Bowen’s for the Axiom A case but several new ideas are needed.
Examples include \(n\)-dimensional dispersive billiards, the induced map for Bunimovich type billiards, hard core billiards studied by Kubo and non-billiard systems such as the Lozi map, geodesic flow on \(S^ 2\) with the Donnay-Burns-Gerber metric, Hamiltonian flows on the two torus with the Donnay-Liverani potentials, etc. In the smooth uniformly hyperbolic case our construction gives rise to finite Markov partitions.

MSC:

37D99 Dynamical systems with hyperbolic behavior
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Smale, Diffeomorphisms with Many Periodic Points pp 63– (1965) · Zbl 0142.41103
[2] Hadamard, J. Math. Pure Appl. 4 pp 27– (1898)
[3] DOI: 10.2307/2370306 · JFM 48.0786.05 · doi:10.2307/2370306
[4] Burns, Ergod. Th. & Dynam. Sys. 9 pp 27– (1989)
[5] Bunimovich, Usp. Math. Nauk 45 pp 97– (1990)
[6] DOI: 10.1007/BF01209400 · doi:10.1007/BF01209400
[7] DOI: 10.1007/BF01942372 · Zbl 0453.60098 · doi:10.1007/BF01942372
[8] DOI: 10.1007/BF02096936 · Zbl 0704.58029 · doi:10.1007/BF02096936
[9] Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms 470 (1975) · Zbl 0308.28010 · doi:10.1007/BFb0081279
[10] DOI: 10.2307/2373370 · Zbl 0208.25901 · doi:10.2307/2373370
[11] DOI: 10.1073/pnas.57.6.1573 · Zbl 0177.08002 · doi:10.1073/pnas.57.6.1573
[12] Misiurewicz, Int. Conf. New York 357 pp 348– (1979)
[13] DOI: 10.1007/BF01033077 · Zbl 0647.58028 · doi:10.1007/BF01033077
[14] Kubo, Nagoya Math. J. 61 pp 1– (1976) · Zbl 0348.58008 · doi:10.1017/S0027763000017281
[15] Katok, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities 1222 (1986) · doi:10.1007/BFb0099031
[16] Katok, Amer. Math. Soc. Transl. 116 pp 43– (1981) · Zbl 0598.58031 · doi:10.1090/trans2/116/02
[17] Gallavotti, Lectures on the Billiards 38 pp 236– (1975)
[18] DOI: 10.1007/BF02098044 · Zbl 0719.58022 · doi:10.1007/BF02098044
[19] Donnay, Part II: Ergodicity; Dynamical Systems Proc; Univ. of Maryland 1986?87 1342 pp 112– (1988)
[20] Donnay, Ergod. Th. & Dynam. Sys. 8 pp 531– (1988)
[21] Sinai, Fund. Anal Appl. 2 pp 70– (1968) · Zbl 0182.55003 · doi:10.1007/BF01075361
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.