January  2017, 37(1): 105-130. doi: 10.3934/dcds.2017005

Limiting distributions for countable state topological Markov chains with holes

1. 

Department of Mathematics, Fairfield University, Fairfield, CT 06824, USA

2. 

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA

3. 

Department of Mathematics, Fairfield University Fairfield, CT 06824, USA

4. 

Department of Electrical Engineering, The Cooper Union, New York, NY 10003, USA

5. 

Department of Mathematics, Columbia University, New York, NY 10027, USA

Received  January 2016 Revised  June 2016 Published  November 2016

Fund Project: The majority of this paper was completed as part of Fairfield University's REU program, funded by NSF grant DMS 1358454. MD was partially supported by NSF grant DMS 1362420

We study the dynamics of countable state topological Markov chains with holes, where the hole is a countable union of 1-cylinders. For a large class of positive recurrent potentials and under natural assumptions on the surviving dynamics, we prove the existence of a limiting conditionally invariant distribution, which is the unique limit of regular densities under the renormalized dynamics conditioned on non-escape. We also prove the existence of a Gibbs measure on the survivor set, the set of points that never enter the hole, which is an equilibrium measure for the punctured potential of the open system. We prove that the Gurevic pressure on the survivor set equals the exponential escape rate from the open system. These results extend to the non-compact setting results previously available for finite state topological Markov chains.

Citation: Mark F. Demers, Christopher J. Ianzano, Philip Mayer, Peter Morfe, Elizabeth C. Yoo. Limiting distributions for countable state topological Markov chains with holes. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 105-130. doi: 10.3934/dcds.2017005
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, American Mathematical Society, 1997,284 pp.Google Scholar

[2]

E. G. AltmannJ. S. E. Portela and T. Tél, Leaking chaotic systems, Rev. Mod. Phys., 85 (2013), 869-918. doi: 10.1103/RevModPhys.85.869. Google Scholar

[3]

N. N. Čencova, A natural invariant measure on Smale's horseshoe, Soviet Math. Dokl., 256 (1981), 294-298. Google Scholar

[4]

N. Chernov and R. Markarian, Ergodic properties of Anosov maps with rectangular holes, Bol. Soc. Bras. Mat., 28 (1997), 271-314. doi: 10.1007/BF01233395. Google Scholar

[5]

N. Chernov and R. Markarian, Anosov maps with rectangular holes. Nonergodic cases, Bol. Soc. Bras. Mat., 28 (1997), 315-342. doi: 10.1007/BF01233396. Google Scholar

[6]

N. ChernovR. Markarian and S. Troubetskoy, Conditionally invariant measures for Anosov maps with small holes, Ergod. Th. and Dynam. Sys., 18 (1998), 1049-1073. doi: 10.1017/S0143385798117492. Google Scholar

[7]

N. ChernovR. Markarian and S. Troubetskoy, Invariant measures for Anosov maps with small holes, Ergod. Th. and Dynam. Sys., 20 (2000), 1007-1044. doi: 10.1017/S0143385700000560. Google Scholar

[8]

H. van den Bedem and N. Chernov, Expanding maps of an interval with holes, Ergod. Th. and Dynam. Sys., 22 (2002), 637-654. doi: 10.1017/S0143385702000329. Google Scholar

[9]

P. Collet, S. Martínez and J. San Martín, Quasi-Stationary Distributions Probability and Its Applications, Springer-Verlag: Berlin Heidelberg, 2013,280 pp. doi: 10.1007/978-3-642-33131-2. Google Scholar

[10]

P. ColletS. Martínez and B. Schmitt, The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems, Nonlinearity, 7 (1994), 1437-1443. doi: 10.1088/0951-7715/7/5/010. Google Scholar

[11]

P. ColletS. Martínez and B. Schmitt, The Yorke-Pianigiani measure for topological Markov chains, Israel J. of Math., 97 (1997), 61-70. doi: 10.1007/BF02774026. Google Scholar

[12]

M. F. Demers, Markov extensions for dynamical systems with holes: An application to expanding maps of the interval, Israel J. of Math., 146 (2005), 189-221. doi: 10.1007/BF02773533. Google Scholar

[13]

M. F. Demers, Markov extensions and conditionally invariant measures for certain logistic maps with small holes, Ergod. Th. and Dynam. Sys., 25 (2005), 1139-1171. doi: 10.1017/S0143385704000963. Google Scholar

[14]

M. F. Demers, Dispersing billiards with small holes, in Ergodic theory, open dynamics and coherent structures, Springer Proceedings in Mathematics, 70 (2014), 137-170. doi: 10.1007/978-1-4939-0419-8_8. Google Scholar

[15]

M. F. Demers, Escape rates and physical measures for the infinite horizon Lorentz gas with holes, Dynamical Systems: An International Journal, 28 (2013), 393-422. doi: 10.1080/14689367.2013.814946. Google Scholar

[16]

H. BruinM. F. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes, Ergod. Th. and Dynam. Sys., 3 (2010), 687-728. doi: 10.1017/S0143385709000200. Google Scholar

[17]

M. F. Demers and B. Fernandez, Escape rates and singular limiting distributions for intermittent maps with holes, Trans. Amer. Math. Soc., 368 (2016), 4907-4932. doi: 10.1090/tran/6481. Google Scholar

[18]

M. F. DemersP. Wright and L.-S. Young, Escape rates and physically relevant measures for billiards with small holes, Comm. Math. Phys., 294 (2010), 353-388. doi: 10.1007/s00220-009-0941-y. Google Scholar

[19]

M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397. doi: 10.1088/0951-7715/19/2/008. Google Scholar

[20]

C. P. Dettmann and O. Georgiou, Survival probability for the stadium billiard, Physica D, 238 (2009), 2395-2403. doi: 10.1016/j.physd.2009.09.019. Google Scholar

[21]

C. P. Dettmann and M. R. Rahman, Survival probability for open spherical billiards, Chaos 24 (2014), 043130, 15 pp. doi: 10.1063/1.4900776. Google Scholar

[22]

P. A. FerrariH. KestenS. Martínez and P. Picco, Existence of quasi-stationary distributions. A renewal dynamical approach, Annals of Prob., 23 (1995), 501-521. doi: 10.1214/aop/1176988277. Google Scholar

[23]

H. Hennion, Sur un théoréme spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348. Google Scholar

[24]

G. Keller, Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc., 314 (1989), 433-497. doi: 10.1090/S0002-9947-1989-1005524-4. Google Scholar

[25]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. Google Scholar

[26]

C. Liverani and V. Maume-Deschamps, 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 (2003), 385-412. doi: 10.1016/S0246-0203(02)00005-5. Google Scholar

[27]

A. Lopes and R. Markarian, Open Billiards: Cantor sets, invariant and conditionally invariant probabilities, SIAM J. Appl. Math., 56 (1996), 651-680. doi: 10.1137/S0036139995279433. Google Scholar

[28]

R. D. Maudlin and M. Urbanski, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math., 125 (2001), 651-680. doi: 10.1007/BF02773377. Google Scholar

[29]

K. R. Parthasarathy, Introduction to Probability and Measure Springer-Verlag, 1977. Google Scholar

[30]

G. Pianigiani and J. A. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos, Trans. Amer. Math. Soc., 252 (1979), 351-366. doi: 10.2307/1998093. Google Scholar

[31]

O. Sarig, Thermodynamical formalism for countable Markov shifts, Ergodic Theory Dyn. Syst., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820. Google Scholar

[32]

O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc., 131 (2003), 1751-1758. doi: 10.1090/S0002-9939-03-06927-2. Google Scholar

[33]

D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math., 13 (1962), 7-28. doi: 10.1093/qmath/13.1.7. Google Scholar

[34]

T. Yarmola, Sub-exponential mixing of random billiards driven by thermostats, Nonlinearity, 26 (2013), 1825-1837. doi: 10.1088/0951-7715/26/7/1825. Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, American Mathematical Society, 1997,284 pp.Google Scholar

[2]

E. G. AltmannJ. S. E. Portela and T. Tél, Leaking chaotic systems, Rev. Mod. Phys., 85 (2013), 869-918. doi: 10.1103/RevModPhys.85.869. Google Scholar

[3]

N. N. Čencova, A natural invariant measure on Smale's horseshoe, Soviet Math. Dokl., 256 (1981), 294-298. Google Scholar

[4]

N. Chernov and R. Markarian, Ergodic properties of Anosov maps with rectangular holes, Bol. Soc. Bras. Mat., 28 (1997), 271-314. doi: 10.1007/BF01233395. Google Scholar

[5]

N. Chernov and R. Markarian, Anosov maps with rectangular holes. Nonergodic cases, Bol. Soc. Bras. Mat., 28 (1997), 315-342. doi: 10.1007/BF01233396. Google Scholar

[6]

N. ChernovR. Markarian and S. Troubetskoy, Conditionally invariant measures for Anosov maps with small holes, Ergod. Th. and Dynam. Sys., 18 (1998), 1049-1073. doi: 10.1017/S0143385798117492. Google Scholar

[7]

N. ChernovR. Markarian and S. Troubetskoy, Invariant measures for Anosov maps with small holes, Ergod. Th. and Dynam. Sys., 20 (2000), 1007-1044. doi: 10.1017/S0143385700000560. Google Scholar

[8]

H. van den Bedem and N. Chernov, Expanding maps of an interval with holes, Ergod. Th. and Dynam. Sys., 22 (2002), 637-654. doi: 10.1017/S0143385702000329. Google Scholar

[9]

P. Collet, S. Martínez and J. San Martín, Quasi-Stationary Distributions Probability and Its Applications, Springer-Verlag: Berlin Heidelberg, 2013,280 pp. doi: 10.1007/978-3-642-33131-2. Google Scholar

[10]

P. ColletS. Martínez and B. Schmitt, The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems, Nonlinearity, 7 (1994), 1437-1443. doi: 10.1088/0951-7715/7/5/010. Google Scholar

[11]

P. ColletS. Martínez and B. Schmitt, The Yorke-Pianigiani measure for topological Markov chains, Israel J. of Math., 97 (1997), 61-70. doi: 10.1007/BF02774026. Google Scholar

[12]

M. F. Demers, Markov extensions for dynamical systems with holes: An application to expanding maps of the interval, Israel J. of Math., 146 (2005), 189-221. doi: 10.1007/BF02773533. Google Scholar

[13]

M. F. Demers, Markov extensions and conditionally invariant measures for certain logistic maps with small holes, Ergod. Th. and Dynam. Sys., 25 (2005), 1139-1171. doi: 10.1017/S0143385704000963. Google Scholar

[14]

M. F. Demers, Dispersing billiards with small holes, in Ergodic theory, open dynamics and coherent structures, Springer Proceedings in Mathematics, 70 (2014), 137-170. doi: 10.1007/978-1-4939-0419-8_8. Google Scholar

[15]

M. F. Demers, Escape rates and physical measures for the infinite horizon Lorentz gas with holes, Dynamical Systems: An International Journal, 28 (2013), 393-422. doi: 10.1080/14689367.2013.814946. Google Scholar

[16]

H. BruinM. F. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes, Ergod. Th. and Dynam. Sys., 3 (2010), 687-728. doi: 10.1017/S0143385709000200. Google Scholar

[17]

M. F. Demers and B. Fernandez, Escape rates and singular limiting distributions for intermittent maps with holes, Trans. Amer. Math. Soc., 368 (2016), 4907-4932. doi: 10.1090/tran/6481. Google Scholar

[18]

M. F. DemersP. Wright and L.-S. Young, Escape rates and physically relevant measures for billiards with small holes, Comm. Math. Phys., 294 (2010), 353-388. doi: 10.1007/s00220-009-0941-y. Google Scholar

[19]

M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397. doi: 10.1088/0951-7715/19/2/008. Google Scholar

[20]

C. P. Dettmann and O. Georgiou, Survival probability for the stadium billiard, Physica D, 238 (2009), 2395-2403. doi: 10.1016/j.physd.2009.09.019. Google Scholar

[21]

C. P. Dettmann and M. R. Rahman, Survival probability for open spherical billiards, Chaos 24 (2014), 043130, 15 pp. doi: 10.1063/1.4900776. Google Scholar

[22]

P. A. FerrariH. KestenS. Martínez and P. Picco, Existence of quasi-stationary distributions. A renewal dynamical approach, Annals of Prob., 23 (1995), 501-521. doi: 10.1214/aop/1176988277. Google Scholar

[23]

H. Hennion, Sur un théoréme spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348. Google Scholar

[24]

G. Keller, Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc., 314 (1989), 433-497. doi: 10.1090/S0002-9947-1989-1005524-4. Google Scholar

[25]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. Google Scholar

[26]

C. Liverani and V. Maume-Deschamps, 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 (2003), 385-412. doi: 10.1016/S0246-0203(02)00005-5. Google Scholar

[27]

A. Lopes and R. Markarian, Open Billiards: Cantor sets, invariant and conditionally invariant probabilities, SIAM J. Appl. Math., 56 (1996), 651-680. doi: 10.1137/S0036139995279433. Google Scholar

[28]

R. D. Maudlin and M. Urbanski, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math., 125 (2001), 651-680. doi: 10.1007/BF02773377. Google Scholar

[29]

K. R. Parthasarathy, Introduction to Probability and Measure Springer-Verlag, 1977. Google Scholar

[30]

G. Pianigiani and J. A. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos, Trans. Amer. Math. Soc., 252 (1979), 351-366. doi: 10.2307/1998093. Google Scholar

[31]

O. Sarig, Thermodynamical formalism for countable Markov shifts, Ergodic Theory Dyn. Syst., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820. Google Scholar

[32]

O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc., 131 (2003), 1751-1758. doi: 10.1090/S0002-9939-03-06927-2. Google Scholar

[33]

D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math., 13 (1962), 7-28. doi: 10.1093/qmath/13.1.7. Google Scholar

[34]

T. Yarmola, Sub-exponential mixing of random billiards driven by thermostats, Nonlinearity, 26 (2013), 1825-1837. doi: 10.1088/0951-7715/26/7/1825. Google Scholar

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