Abstract
We study Anosov diffeomorphisms on manifolds in which some ‘holes’ are cut. The points that are mapped into our holes will disappear and never return. We study the case where the holes are rectangles of a Markov partition. Such maps with holes generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of our map is a Cantor-like set we call arepeller. In our previous paper, we assumed that the map restricted to the remaining rectangles of the Markov partition is topologically mixing. Under this assumption we constructed invariant and conditionally invariant measures on the sets of nonwandering points. Here we relax the mixing assumption and extend our results to nonmixing and nonergodic cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes Math.470, Springer-Verlag, Berlin, 1975.
N.N. Čencova,A natural invariant measure on Smale's horseshoe, Soviet Math. Dokl.23 (1981), 87–91.
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.
N. Chernov and R. Markarian,Ergodic properties of Anosov maps with rectangular holes, Bol. Soc. Bras. Mat., Vol. 28,2, (1997), pp. 271–314.
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.
P. Gaspard and J.R. Dorfman,Chaotic scattering theory, thermodynamic formalism, and transport coefficients, preprint, 1995.
P. Gaspard and G. Nicolis,Transport properties, Lyapunov exponents, and entropy per unit time, Phys. Rev. Lett.65 (1990), 1693–1696.
P. Gaspard and S. Rice,Scattering from a classically chaotic repellor, J. Chem. Phys.90 (1989), 2225–2241.
A. Lopes and R. Markarian,Open billiards: Cantor sets, invariant and conditionally invariant probabilities, SIAM J. on Applied Math.56 (1996), 651–680.
G. Pianigiani and J. Yorke,Expanding maps on sets which are almost invariant: decay and chaos, Trans. Amer. Math. Soc.252 (1979), 351–366.
D. Ruelle,A measure associated with Axiom A attractors, Amer. J. Math.98 (1976), 619–654.
Ya.G. Sinai,Markov partitions and C-diffeomorphisms, Funct. Anal. Its Appl.2 (1968), 61–82.
Ya.G. Sinai,Gibbs measures in ergodic theory, Russ. Math. Surveys27 (1972), 21–69.
Author information
Authors and Affiliations
Additional information
To the memory of Ricardo Mañé
Partially supported by NSF grant DMS-9401417.
Partially supported by CONICYT and CSIC, Univ. de la Republica (Uruguay).
About this article
Cite this article
Chernov, N., Markarian, R. Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat 28, 315–342 (1997). https://doi.org/10.1007/BF01233396
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01233396