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 and Continuous Dynamical Systems, 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

show all references

References:
[1]

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

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[29]

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

[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.

[31]

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

[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.

[33]

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

[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.

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