Markov extensions and conditionally invariant measures for certain logistic maps with small holes

Ergodic Theory and Dynamical Systems

Volumn 25, Issue 4, 2005, Pages 1139-1171

  Demers, Mark F ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23444450388     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0143385704000963     Document Type: Article

References (26)

1. 2 on (-1, 1). Ann. Math. 122 (1985), 1-25.
2. [BCh] H. van den Bedem and N. Chernov. Expanding maps of an interval with holes. Ergod. Th. & Dynam. Sys. 22 (2002), 637-654.
3. [BK] V. Baladi and G. Keller. Zeta functions and transfer operators for piecewise monotonic transformations. Comm. Math. Phys. 127 (1990), 459-477.
4. [BY] M. Benedicks and L.-S. Young. Markov extensions and decay of correlations for certain Hénon maps. Astérisque 261 (2000), xi, 13-56.
5. [C1] N. N. Cencova. A natural invariant measure on Smale's horseshoe. Soviet Math. Dokl. 23 (1981), 87-91.
6. [C2] N. N. Cencova. Statistical properties of smooth Smale horseshoes. Mathematical Problems of Statistical Mechanics and Dynamics. Ed. R. L. Dobrushin. Reidel, Dordrecht, 1986, pp. 199-256.
7. [Ch1] N. Chernov. Statistical properties of piecewise smooth hyperbolic systems in high dimensions. Discr. Cont. Dynam. Syst. 5 (1999), 425-448.
8. [Ch2] N. Chernov. Decay of correlations and dispersing billiards. J. Stat. Phys. 94 (1999), 513-556.
9. [Ch3] N. Chernov. Sinai billiards under small external forces. Ann. Henri Poincaré 2 (2001), 197-236.
10. [CM1] N. Chernov and R. Markarian. Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28 (1997), 271-314.
11. [CM2] N. Chernov and R. Markarian. Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28 (1997), 315-342.
12. [CMS1] P. Collet, S. Martinez and B. Schmitt. The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7 (1994), 1437-1443.
13. [CMS2] P. Collet, S. Martinez and B. Schmitt. Quasi-stationary distribution and Gibbs measure of expanding systems. Instabilities and Nonequilibrium Structures V. Eds. E. Tirapegui and W. Zeller. Kluwer, Dordrecht, 1996, pp. 205-219.
14. [CMT1] N. Chernov, R. Markarian and S. Troubetskoy. Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 18 (1998), 1049-1073.
15. [CMT2] N. Chernov, R. Markarian and S. Troubetskoy. Invariant measures for Anosov maps with small holes. Ergod. Th. & Dynam. Sys. 20 (2000), 1007-1044.
16. [D] M. Demers. Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. Israel J. Math. 146 (2005), 189-221.
17. [KL] G. Keller and C. Liverani. Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 141-152.
18. [LiM] C. Liverani and V. Maume-Deschamps. Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Ann. Henri Poincaré Prob. Stat. 39 (2003), 385-412.
19. [LM] A. Lopes and R. Markarian. Open billiards: Cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56 (1996), 651-680.
20. [PY] G. Pianigiani and J. Yorke. Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351-366.
21. [R] P. A. Richardson, Jr. Natural measures on the unstable and invariant manifolds of open billiard dynamical systems. Doctoral Dissertation, Department of Mathematics, University of North Texas, 1999.
22. [WY1] Q. Wang and L.-S. Young. Strange attractors with one direction of instability. Comm. Math. Phys. 218 (2001), 1-97.
23. [WY2] Q. Wang and L.-S. Young. Nonuniformly expanding one-dimensional maps. Preprint, 2005.
24. [Y1] L.-S. Young. Decay of correlations for certain quadratic maps. Comm. Math. Phys. 146 (1992), 123-138.
25. [Y2] L.-S. Young. Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147 (1998), 585-650.
26. [Y3] L.-S. Young. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153-188.