Ergodic properties of Anosov maps with rectangular holes

  • N. Chernov
  • R. Markarian


We study Anosov diffeomorphisms on manifolds in which some ‘holes’ are cut. The points that are mapped into those holes disappear and never return. The holes studied here are rectangles of a Markov partition. Such maps generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of a map of this kind is a Cantor-like set calledrepeller. We construct invariant and conditionally invariant measures on the sets of nonwandering points. Then we establish ergodic, statistical, and fractal properties of those measures.


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  1. [1]
    D.V. Anosov and Ya.G. Sinai,Some smooth ergodic systems, Russ. Math. Surveys22 (1967), 103–167.Google Scholar
  2. [2]
    R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes Math.470, Springer-Verlag, Berlin, 1975.Google Scholar
  3. [3]
    N.N. Čencova,A natural invariant measure on Smale's horseshoe, Soviet Math. Dokl.23 (1981), 87–91.Google Scholar
  4. [4]
    N.N. Čencova,Statistical properties of smooth Smale horseshoes, in:Mathematical Problems of Statistical Mechanics and Dynamics, R. L. Dobrushin, Editor, pp. 199–256, Reidel, Dordrecht, 1986.Google Scholar
  5. [5]
    N.I. Chernov, G.L. Eyink, J.L. Lebowitz and Ya.G. Sinai,Steady-state electrical conduction in the periodic Lorentz gas, Comm. Math. Phys.154 (1993), 569–601.Google Scholar
  6. [6]
    P. Collet, S. Martinez and B. Schmitt,The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems, Nonlinearity7 (1994), 1437–1443.Google Scholar
  7. [7]
    P. Collet, S. Martinez and B. Schmitt,The Pianigiani-Yorke measure for topological Markov chains, manuscript, 1994.Google Scholar
  8. [8]
    P. Ferrari, H. Kesten, S. Martinez and P. Picco,Existence of quasi stationary distribution. A renewal dynamical approach, to appear in Annals Probab.Google Scholar
  9. [9]
    P. Gaspard and F. Baras,Chaotic scattering and diffusion in the Lorentz gas, Phys. Rev. E51 (1995), 5332–5352.Google Scholar
  10. [10]
    P. Gaspard and J.R. Dorfman,Chaotic scattering theory, thermodynamic formalism, and transport coefficients, preprint, 1995.Google Scholar
  11. [11]
    P. Gaspard and G. Nicolis,Transport properties, Lyapunov exponents, and entropy per unit time, Phys. Rev. Lett.65 (1990), 1693–1696.Google Scholar
  12. [12]
    P. Gaspard and S. Rice,Scattering from a classically chaotic repellor, J. Chem. Phys.90 (1989), 2225–2241.Google Scholar
  13. [13]
    Y. Guivarc'h and J. Hardy,Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincaré,24 (1988), 73–98.Google Scholar
  14. [14]
    M. Keane and M. Mori, Dynamical systems on the Cantor sets associated with piecewise linear transformations, manuscript.Google Scholar
  15. [15]
    O. Legrand and D. Sornette,Coarse-grained properties of the chaotic trajectories in the stadium, Phys. D44 (1990), 229–247.Google Scholar
  16. [16]
    A. Lopes and R. Markarian,Open billiards: Cantor sets, invariant and conditionally invariant probabilities, SIAM J.of Applied Math,56 (1996), 651–680.Google Scholar
  17. [17]
    R. Mañé,Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987.Google Scholar
  18. [18]
    A. Manning,A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergod. Th. & Dynam. Sys.1 (1981), 451–459.Google Scholar
  19. [19]
    S. Martinez and M.E. Vares,Markov chain associated to the minimal Q.S.D. of birth-rate chains, to appear in J. Applied Probab.Google Scholar
  20. [20]
    L. Mendoza,The entropy of C 2 surface diffeormorphisms in terms of Hausdorff dimension and a Lyapunov exponent, Ergod. Th. & Dynam. Sys.5 (1985), 273–283.Google Scholar
  21. [21]
    Z. Nitecki,Differentiable Dynamics, MIT Press, Cambridge, Mass., 1971.Google Scholar
  22. [22]
    G. Pianigiani and J. Yorke,Expanding maps on sets which are almost invariant: decay and chaos, Trans. Amer. Math. Soc.252 (1979), 351–366.Google Scholar
  23. [23]
    Ya.G. Sinai,Markov partitions and C-diffeomorphisms, Funct. Anal. Its Appl.2 (1968), 61–82.Google Scholar
  24. [24]
    Ya.G. Sinai,Gibbs measures in ergodic theory, Russ. Math. Surveys27 (1972), 21–69.Google Scholar
  25. [25]
    L.-S. Young,Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys.2 (1982), 109–124.Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 1997

Authors and Affiliations

  • N. Chernov
    • 1
  • R. Markarian
    • 2
  1. 1.Department of MathematicsUniversity of Alabama in BirminghamBirminghamUSA
  2. 2.Instituto de Matemática y Estadística “Prof. Ing. Rafael Laguardia”Facultad de Ingenieríanewline Universidad de la RepúblicaMontevideoUruguay

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