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Escape rates and physically relevant measures for billiards with small holes. (English) Zbl 1225.37051
This paper focuses on the subject of leaky dynamical systems, where the leaks in the system are caused by possible holes in the phase space. The system starts with an initial probability distribution and as the mass leaks out of the system by time, the main concern is the distribution of the remaining mass at that instant. This is also related to the rate of escape of the mass from the system. The authors give a rigorous and detailed study of these distribution measures on billiard systems with small holes. Such measures turn out to be physically relevant and correspond to the two dimensional periodic Lorentz gas model. For these systems, they prove the existence of a common rate of escape and a common limiting distribution for a large class of initial distributions including the ones arising from the Liouville measure. In this respect, the authors construct Markov tower extensions that are compatible with holes in the system. These towers are related to sufficiently strong hyperbolic properties. The actual construction of these towers is quite elaborate and forms the heart of this paper. The information obtained is then interpreted for the billiard system, proving weak convergence of all the relevant distribution measures.

37D50Hyperbolic systems with singularities (billiards, etc.)
37C40Smooth ergodic theory, invariant measures
37N20Dynamical systems in other branches of physics
Full Text: DOI arXiv
[1] Baladi V., Keller G.: Zeta functions and transfer operators for piecewise monotonic transformations. Commun. Math. Phys. 127, 459--477 (1990) · Zbl 0703.58048 · doi:10.1007/BF02104498
[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. · Zbl 1202.37058
[3] Bunimovich L.A., Sinaĭ Ya.G., Chernov N.I.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45(3), 105--152 (1990) · Zbl 0721.58036 · doi:10.1070/RM1990v045n03ABEH002355
[4] Bunimovich L.A., Sinaĭ Ya.G., Chernov N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4), 47--106 (1991) · Zbl 0780.58029 · doi:10.1070/RM1991v046n04ABEH002827
[5] Buzzi J.: Markov extensions for multidimensional dynamical systems. Israel J. of Math. 112, 357--380 (1999) · Zbl 0988.37012 · doi:10.1007/BF02773488
[6] Cenvoca N.N.: A natural invariant measure on Smale’s horseshoe. Soviet Math. Dokl. 23, 87--91 (1981) · Zbl 0475.58013
[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) · Zbl 0893.58035 · doi:10.1007/BF01233395
[9] Chernov N., Markarian R.: Anosov maps with rectangular holes. Nonergodic cases. Bol. Soc. Bras. Mat. 28, 315--342 (1997) · Zbl 0893.58036 · doi:10.1007/BF01233396
[10] Chernov, N., Markarian, R.: Chaotic Billiards. Number 127 in Mathematical Surveys and Monographs, Providence, RI: Amer. Math. Soc., 2006 · Zbl 1101.37001
[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) · Zbl 0982.37011 · doi:10.1017/S0143385798117492
[12] Chernov N., Markarian R., Troubetzkoy S.: Invariant measures for Anosov maps with small holes. Ergod. Th. and Dynam. Sys. 20, 1007--1044 (2000) · Zbl 0963.37030 · doi:10.1017/S0143385700000560
[13] Collet P., Martínez S., Maume-Deschamps V.: On the existence of conditionally invariant probability measures in dynamical systems. Nonlinearity 13, 1263--1274 (2000) · Zbl 0974.37003 · doi:10.1088/0951-7715/13/4/315
[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) · Zbl 0806.58037 · doi:10.1088/0951-7715/7/5/010
[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 · Zbl 0897.60097
[16] Collet P., Martínez S., Schmitt B.: The Pianigiani-Yorke measure for topological Markov chains. Israel J. Math. 97, 61--70 (1997) · Zbl 0902.58016 · doi:10.1007/BF02774026
[17] Chernov N., van dem Bedem H.: Expanding maps of an interval with holes. Ergod. Th. and Dynam. Sys. 22, 637--654 (2002) · Zbl 1022.37024
[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) · Zbl 1079.37031 · doi:10.1007/BF02773533
[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) · Zbl 1098.37035 · doi:10.1017/S0143385704000963
[20] Demers M., Liverani C.: Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9), 4777--4814 (2008) · Zbl 1153.37019 · doi:10.1090/S0002-9947-08-04464-4
[21] Demers M., Young L.-S.: Escape rates and conditionally invariant measures. Nonlinearity 19, 377--397 (2006) · Zbl 1134.37322 · doi:10.1088/0951-7715/19/2/008
[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) · Zbl 0827.60061
[23] Halmos, P.R.: Measure Theory. University Series in Higher Mathematics, Princeton, NJ: D. Van Nostrand Co., Inc., 1950, 304 p. · Zbl 0040.16802
[24] Homburg A., Young T.: Intermittency in families of unimodal maps. Ergod. Th. and Dynam. Sys. 22(1), 203--225 (2002) · Zbl 1032.37020 · doi:10.1017/S0143385702000093
[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 · Zbl 0658.58001
[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) · Zbl 1021.37002 · doi:10.1016/S0246-0203(02)00005-5
[27] Lopes A., Markarian R.: Open Billiards: Cantor sets, invariant and conditionally invariant probabilities. SIAM J. Appl. Math. 56, 651--680 (1996) · Zbl 0852.58056 · doi:10.1137/S0036139995279433
[28] Pianigiani G., Yorke J.: Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252, 351--366 (1979) · Zbl 0417.28010
[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] Sinaĭ Ya.G.: Dynamical systems with elastic collisions. Ergodic properties of dispersing billiards. Usp. Mat. Nauk 25(2), 141--192 (1970) · Zbl 0252.58005
[31] Vere-Jones D.: Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13, 7--28 (1962) · Zbl 0104.11805 · doi:10.1093/qmath/13.1.7
[32] Wojtkowski M.: Invariant families of cones and Lyapunov exponents. Ergod. Th. Dynam. Sys. 5(1), 145--161 (1985) · Zbl 0578.58033 · doi:10.1017/S0143385700002807
[33] Young L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Annals of Math. 147(3), 585--650 (1998) · Zbl 0945.37009 · doi:10.2307/120960