×

Ergodic properties of Anosov maps with rectangular holes. (English) Zbl 0893.58035

Let \(T:M'\to M'\) be a topologically transitive Anosov diffeomorphism of class \(C^{1+\varepsilon}\) on a compact Riemannian manifold \(M'\). Let \({\mathcal R} =(R_1, \dots, R_{I'}) \) be a Markov partition on \(M'\). Let \(I<I'\), \(H=\bigcup^{I'}_{i=I+1} (\text{int} R_i)\), \(M=M' \setminus H\).
The dynamics of \(T\) on \(M\), thinking of \(H\) as a ‘hole’ into which some points are mapped and disappear, is studied. It is assumed that the symbolic dynamics generated by the partition \(\{R_1, \dots, R_I\}\) is a topologically mixing subshift of finite type. Denote \(M_n= \bigcap^n_{i=0} T^iM\), \(M_{-n} =\bigcap^n_{i=0} T^{-i}M\), \(M_+ =\bigcap_{n\geq 1} M_n\), \(M_-= \bigcap_{n\geq 1} M_{-n}\), \(\Omega= M_+ \bigcap M_-\). \({\mathcal U}' =\bigvee^\infty_{n=0} T^n {\mathcal R}'\), \({\mathcal S}' =\bigvee^\infty_{n=0} T^{-n} {\mathcal R}'\) are the partitions of \(M'\) into unstable and stable fibers, respectively. The restrictions of \({\mathcal U}'\) to \(M\) \((M_+)\) are denoted by \({\mathcal U}\) \(({\mathcal U}_+)\), respectively. A conditionally invariant measure is defined.
Some of the main results of the paper are the following. There is a unique conditionally invariant family of probability measures \(\nu^u_U\), on fibers \(U\in {\mathcal U}_+\). Denote by \({\mathcal M}^n_+\) the class of Borel measures supported on fibers \(U\in {\mathcal U}_+\) coinciding with \(\nu^n_U\). The map \(T\) has a unique conditionally invariant measure \(\mu_+ \in {\mathcal M}^n_+\). The sequence of measures \(T_*^{-n} \mu_+\) \((T_* \mu= \mu T^{-1})\) weakly converges to a probability measure \(\eta+\) supported on \(\Omega\). The measure \(\nu_+\) is invariant, ergodic, mixing, \(K\)- mixing and Bernoulli. Its correlations decay exponentially fast and it satisfies the central limit theorems. \(\eta_+\) is Gibbs’ measure. The variational principle for \(\eta_+\) holds.

MSC:

37A99 Ergodic theory
37E99 Low-dimensional dynamical systems
37D99 Dynamical systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] D.V. Anosov and Ya.G. Sinai,Some smooth ergodic systems, Russ. Math. Surveys22 (1967), 103-167. · Zbl 0177.42002
[2] R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes Math.470, Springer-Verlag, Berlin, 1975. · Zbl 0308.28010
[3] N.N. ?encova,A natural invariant measure on Smale’s horseshoe, Soviet Math. Dokl.23 (1981), 87-91. · Zbl 0475.58013
[4] N.N. ?encova,Statistical properties of smooth Smale horseshoes, in:Mathematical Problems of Statistical Mechanics and Dynamics, R. L. Dobrushin, Editor, pp. 199-256, Reidel, Dordrecht, 1986.
[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. · Zbl 0780.58050
[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. · Zbl 0806.58037
[7] P. Collet, S. Martinez and B. Schmitt,The Pianigiani-Yorke measure for topological Markov chains, manuscript, 1994.
[8] P. Ferrari, H. Kesten, S. Martinez and P. Picco,Existence of quasi stationary distribution. A renewal dynamical approach, to appear in Annals Probab. · Zbl 0827.60061
[9] P. Gaspard and F. Baras,Chaotic scattering and diffusion in the Lorentz gas, Phys. Rev. E51 (1995), 5332-5352.
[10] P. Gaspard and J.R. Dorfman,Chaotic scattering theory, thermodynamic formalism, and transport coefficients, preprint, 1995.
[11] P. Gaspard and G. Nicolis,Transport properties, Lyapunov exponents, and entropy per unit time, Phys. Rev. Lett.65 (1990), 1693-1696. · Zbl 1050.82547
[12] P. Gaspard and S. Rice,Scattering from a classically chaotic repellor, J. Chem. Phys.90 (1989), 2225-2241.
[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.
[14] M. Keane and M. Mori, Dynamical systems on the Cantor sets associated with piecewise linear transformations, manuscript.
[15] O. Legrand and D. Sornette,Coarse-grained properties of the chaotic trajectories in the stadium, Phys. D44 (1990), 229-247. · Zbl 0711.58022
[16] A. Lopes and R. Markarian,Open billiards: Cantor sets, invariant and conditionally invariant probabilities, SIAM J.of Applied Math,56 (1996), 651-680. · Zbl 0852.58056
[17] R. Mañé,Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987.
[18] A. Manning,A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergod. Th. & Dynam. Sys.1 (1981), 451-459. · Zbl 0487.58011
[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.
[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. · Zbl 0551.28008
[21] Z. Nitecki,Differentiable Dynamics, MIT Press, Cambridge, Mass., 1971.
[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. · Zbl 0417.28010
[23] Ya.G. Sinai,Markov partitions and C-diffeomorphisms, Funct. Anal. Its Appl.2 (1968), 61-82. · Zbl 0182.55003
[24] Ya.G. Sinai,Gibbs measures in ergodic theory, Russ. Math. Surveys27 (1972), 21-69. · Zbl 0246.28008
[25] L.-S. Young,Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys.2 (1982), 109-124. · Zbl 0523.58024
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.