Escape rates and physically relevant measures for billiards with small holes

Communications in Mathematical Physics

Volumn 294, Issue 2, 2010, Pages 353-388

  Demers, Mark ^a   Wright, Paul ^b   Young, Lai Sang ^c  

^a FAIRFIELD UNIVERSITY   (United States)

^b University of Maryland   (United States)

^c NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 76349119232     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-009-0941-y     Document Type: Article

References (33)

1. Baladi V., Keller G.: Zeta functions and transfer operators for piecewise monotonic transformations. Commun. Math. Phys. 127, 459-477 (1990).
2. Bruin, H., Demers, M., Melbourne, I.: Convergence properties and an equilibrium principle for certain dynamical systems with holes. To appear in Ergod. Th. and Dynam. Sys.
3. Bunimovich L. A., Sinaǐ Ya. G., Chernov N. I.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45(3), 105-152 (1990).
4. Bunimovich L. A., Sinaǐ Ya. G., Chernov N. I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4), 47-106 (1991).
5. Buzzi J.: Markov extensions for multidimensional dynamical systems. Israel J. of Math. 112, 357-380 (1999).
6. Cenvoca N. N.: A natural invariant measure on Smale's horseshoe. Soviet Math. Dokl. 23, 87-91 (1981).
7. Cenvoca, N. N.: Statistical properties of smooth Smale horseshoes. In: Mathematical Problems of Statistical Mechanics and Dynamics, R. L. Dobrushin, ed. Dordrecht: Reidel, 1986, pp. 199-256.
8. Chernov N., Markarian R.: Ergodic properties of Anosov maps with rectangular holes. Bol. Soc. Bras. Mat. 28, 271-314 (1997).
9. Chernov N., Markarian R.: Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28, 315-342 (1997).
10. Chernov, N., Markarian, R.: Chaotic Billiards. Number 127 in Mathematical Surveys and Monographs, Providence, RI: Amer. Math. Soc., 2006.
11. Chernov N., Markarian R., Troubetzkoy S.: Conditionally invariant measures for Anosov maps with small holes. Ergod. Th. and Dynam. Sys. 18, 1049-1073 (1998).
12. Chernov N., Markarian R., Troubetzkoy S.: Invariant measures for Anosov maps with small holes. Ergod. Th. and Dynam. Sys. 20, 1007-1044 (2000).
13. Collet P., Martínez S., Maume-Deschamps V.: On the existence of conditionally invariant probability measures in dynamical systems. Nonlinearity 13, 1263-1274 (2000).
14. Collet P., Martínez S., Schmitt B.: The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7, 1437-1443 (1994).
15. Collet, P., Martínez, S., Schmitt, B.: Quasi-stationary distribution and Gibbs measure of expanding systems. In: Instabilities and Nonequilibrium Structures. V. E. Tirapegui, W. Zeller, eds. Dordrecht: Kluwer, 1996, pp. 205-219.
16. Collet P., Martínez S., Schmitt B.: The Pianigiani-Yorke measure for topological Markov chains. Israel J. Math. 97, 61-70 (1997).
17. Chernov N., van dem Bedem H.: Expanding maps of an interval with holes. Ergod. Th. and Dynam. Sys. 22, 637-654 (2002).
18. Demers M.: Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval. Israel J. of Math. 146, 189-221 (2005).
19. Demers M.: Markov Extensions and Conditionally Invariant Measures for Certain Logistic Maps with Small Holes. Ergod. Th. and Dynam. Sys. 25(4), 1139-1171 (2005).
20. Demers M., Liverani C.: Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9), 4777-4814 (2008).
21. Demers M., Young L.-S.: Escape rates and conditionally invariant measures. Nonlinearity 19, 377-397 (2006).
22. Ferrari P. A., Kesten H., Martínez S., Picco P.: Existence of quasi-stationary distributions. A renewal dynamical approach. Annals of Prob. 23(2), 501-521 (1995).
23. Halmos, P. R.: Measure Theory. University Series in Higher Mathematics, Princeton, NJ: D. van Nostrand Co., Inc., 1950, 304 p.
24. Homburg A., Young T.: Intermittency in families of unimodal maps. Ergod. Th. and Dynam. Sys. 22(1), 203-225 (2002).
25. Katok, A., Strelcyn, J. M.: Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Volume 1222, Springer Lecture Notes in Math., Berlin-Heidelberg-NewYork: Springer, 1986.
26. Liverani C., Maume-Deschamps V.: Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Annales de l'Institut Henri Poincaré Probability and Statistics 39, 385-412 (2003).
27. Lopes A., Markarian R.: Open Billiards: Cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56, 651-680 (1996).
28. Pianigiani G., Yorke J.: Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252, 351-366 (1979).
29. Richardson, P. A., Jr.: Natural Measures on the Unstable and Invariant Manifolds of Open Billiard Dynamical Systems. Doctoral Dissertation, Department of Mathematics, University of North Texas, 1999.
30. Sinaǐ Ya. G.: Dynamical systems with elastic collisions. Ergodic properties of dispersing billiards. Usp. Mat. Nauk 25(2), 141-192 (1970).
31. Vere-Jones D.: Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13, 7-28 (1962).
32. Wojtkowski M.: Invariant families of cones and Lyapunov exponents. Ergod. Th. Dynam. Sys. 5(1), 145-161 (1985).
33. Young L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Annals of Math. 147(3), 585-650 (1998).