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March  2015, 35(3): 1285-1326. doi: 10.3934/dcds.2015.35.1285

Equilibrium states and invariant measures for random dynamical systems

Ivan Werner 1,

1. 

Udal'tsov Street 55, Apt. 17, Moscow, 119454, Russian Federation

Received  August 2013 Revised  July 2014 Published  October 2014

Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.
Keywords: contractive Markov systems, random dynamical systems, Markov partitions, Equilibrium states, Markov operators..
Mathematics Subject Classification: Primary: 37Hxx, 37H99, 82B26; Secondary: 60G1.
Citation: Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
References:
[1]

A. Baraviera, C. F. Lardizabal, A. O. Lopes and M. Terra Cunha, A dynamical point of view of Quantum Information: Entropy, pressure and Wigner measures,, in Dynamics, 2 (2011), 161.  doi: 10.1007/978-3-642-14788-3_13.  Google Scholar

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M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Invariant measure for Markov processes arising from iterated function systems with place-dependent probabilities,, Ann. Inst. Henri Poincaré, 24 (1988), 367.   Google Scholar

[3]

M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Erratum: Invariant measure for Markov processes arising from iterated function systems with place-dependent probabilities,, Ann. Inst. Henri Poincaré, 25 (1989), 589.   Google Scholar

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V. I. Bogachev, Measure Theory. Vol. I,II., Springer, (2007).  doi: 10.1007/978-3-540-34514-5.  Google Scholar

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M. Denker and M. Urbański, On the existence of conformal measures,, Trans. Am. Math. Soc., 328 (1991), 563.  doi: 10.1090/S0002-9947-1991-1014246-4.  Google Scholar

[6]

H. Föllmer, U. Horst and A. Kirman, Equilibria in financial markets with heterogeneous agents: A probabilistic perspective,, Journal of Mathematical Economics, 41 (2005), 123.  doi: 10.1016/j.jmateco.2004.08.001.  Google Scholar

[7]

K. Horbacz and T. Szarek, Irreducible Markov systems on Polish spaces,, Studia Math., 177 (2006), 285.  doi: 10.4064/sm177-3-7.  Google Scholar

[8]

R. Isaac, Markov processes and unique stationary probability measures,, Pacific J. Math., 12 (1962), 273.  doi: 10.2140/pjm.1962.12.273.  Google Scholar

[9]

A. Johansson, A. Öberg and M. Pollicott, Countable state shifts and uniqueness of g-measures,, Amer. J. Math., 129 (2007), 1501.  doi: 10.1353/ajm.2007.0044.  Google Scholar

[10]

M. Keane, Strongly Mixing $g$-Measures,, Inventiones math., 16 (1972), 309.  doi: 10.1007/BF01425715.  Google Scholar

[11]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques,, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 30 (1974), 185.  doi: 10.1007/BF00533471.  Google Scholar

[12]

O. Sarig, Thermodynamic formalism for countable Markov shifts,, Ergod. Th. & Dynam. Sys., 19 (1999), 1565.  doi: 10.1017/S0143385799146820.  Google Scholar

[13]

O. Sarig, Thermodynamic formalism for null recurrent potentials,, Israel Journal of Mathematics, 121 (2001), 285.  doi: 10.1007/BF02802508.  Google Scholar

[14]

W. Slomczynski, Dynamical Entropy, Markov Operators, and Iterated Function Systems,, Rozprawy Habilitacyjne Uniwersytetu Jagiellońskiego Nr 362, (2003).   Google Scholar

[15]

T. Szarek, Invariant measures for nonexpansive Markov operators on Polish spaces,, Diss. Math., 415 (2003), 1.  doi: 10.4064/dm415-0-1.  Google Scholar

[16]

P. Walters, Ruelle's Operator Theorem and $g$-measures,, Tran. AMS, 214 (1975), 375.  doi: 10.1090/S0002-9947-1975-0412389-8.  Google Scholar

[17]

I. Werner, Contractive Markov systems,, J. London Math. Soc., 71 (2005), 236.  doi: 10.1112/S0024610704006088.  Google Scholar

[18]

I. Werner, Coding map for a contractive Markov system,, Math. Proc. Camb. Phil. Soc., 140 (2006), 333.  doi: 10.1017/S0305004105009072.  Google Scholar

[19]

I. Werner, The generalized Markov measure as an equilibrium state,, Nonlinearity 18 (2005), 18 (2005), 2261.  doi: 10.1088/0951-7715/18/5/019.  Google Scholar

[20]

I. Werner, Dynamically defined measures and equilibrium states,, J. Math. Phys. 52 (2011), 52 (2011).  doi: 10.1063/1.3666020.  Google Scholar

[21]

I. Werner, Erratum: Dynamically defined measures and equilibrium states,, J. Math. Phys. 53 (2012), 53 (2012).  doi: 10.1063/1.4736999.  Google Scholar

[22]

I. Werner, Contractive Markov systems II,, , ().   Google Scholar

[23]

I. Werner, Fundamental Markov systems,, , ().   Google Scholar

show all references

References:
[1]

A. Baraviera, C. F. Lardizabal, A. O. Lopes and M. Terra Cunha, A dynamical point of view of Quantum Information: Entropy, pressure and Wigner measures,, in Dynamics, 2 (2011), 161.  doi: 10.1007/978-3-642-14788-3_13.  Google Scholar

[2]

M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Invariant measure for Markov processes arising from iterated function systems with place-dependent probabilities,, Ann. Inst. Henri Poincaré, 24 (1988), 367.   Google Scholar

[3]

M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Erratum: Invariant measure for Markov processes arising from iterated function systems with place-dependent probabilities,, Ann. Inst. Henri Poincaré, 25 (1989), 589.   Google Scholar

[4]

V. I. Bogachev, Measure Theory. Vol. I,II., Springer, (2007).  doi: 10.1007/978-3-540-34514-5.  Google Scholar

[5]

M. Denker and M. Urbański, On the existence of conformal measures,, Trans. Am. Math. Soc., 328 (1991), 563.  doi: 10.1090/S0002-9947-1991-1014246-4.  Google Scholar

[6]

H. Föllmer, U. Horst and A. Kirman, Equilibria in financial markets with heterogeneous agents: A probabilistic perspective,, Journal of Mathematical Economics, 41 (2005), 123.  doi: 10.1016/j.jmateco.2004.08.001.  Google Scholar

[7]

K. Horbacz and T. Szarek, Irreducible Markov systems on Polish spaces,, Studia Math., 177 (2006), 285.  doi: 10.4064/sm177-3-7.  Google Scholar

[8]

R. Isaac, Markov processes and unique stationary probability measures,, Pacific J. Math., 12 (1962), 273.  doi: 10.2140/pjm.1962.12.273.  Google Scholar

[9]

A. Johansson, A. Öberg and M. Pollicott, Countable state shifts and uniqueness of g-measures,, Amer. J. Math., 129 (2007), 1501.  doi: 10.1353/ajm.2007.0044.  Google Scholar

[10]

M. Keane, Strongly Mixing $g$-Measures,, Inventiones math., 16 (1972), 309.  doi: 10.1007/BF01425715.  Google Scholar

[11]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques,, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 30 (1974), 185.  doi: 10.1007/BF00533471.  Google Scholar

[12]

O. Sarig, Thermodynamic formalism for countable Markov shifts,, Ergod. Th. & Dynam. Sys., 19 (1999), 1565.  doi: 10.1017/S0143385799146820.  Google Scholar

[13]

O. Sarig, Thermodynamic formalism for null recurrent potentials,, Israel Journal of Mathematics, 121 (2001), 285.  doi: 10.1007/BF02802508.  Google Scholar

[14]

W. Slomczynski, Dynamical Entropy, Markov Operators, and Iterated Function Systems,, Rozprawy Habilitacyjne Uniwersytetu Jagiellońskiego Nr 362, (2003).   Google Scholar

[15]

T. Szarek, Invariant measures for nonexpansive Markov operators on Polish spaces,, Diss. Math., 415 (2003), 1.  doi: 10.4064/dm415-0-1.  Google Scholar

[16]

P. Walters, Ruelle's Operator Theorem and $g$-measures,, Tran. AMS, 214 (1975), 375.  doi: 10.1090/S0002-9947-1975-0412389-8.  Google Scholar

[17]

I. Werner, Contractive Markov systems,, J. London Math. Soc., 71 (2005), 236.  doi: 10.1112/S0024610704006088.  Google Scholar

[18]

I. Werner, Coding map for a contractive Markov system,, Math. Proc. Camb. Phil. Soc., 140 (2006), 333.  doi: 10.1017/S0305004105009072.  Google Scholar

[19]

I. Werner, The generalized Markov measure as an equilibrium state,, Nonlinearity 18 (2005), 18 (2005), 2261.  doi: 10.1088/0951-7715/18/5/019.  Google Scholar

[20]

I. Werner, Dynamically defined measures and equilibrium states,, J. Math. Phys. 52 (2011), 52 (2011).  doi: 10.1063/1.3666020.  Google Scholar

[21]

I. Werner, Erratum: Dynamically defined measures and equilibrium states,, J. Math. Phys. 53 (2012), 53 (2012).  doi: 10.1063/1.4736999.  Google Scholar

[22]

I. Werner, Contractive Markov systems II,, , ().   Google Scholar

[23]

I. Werner, Fundamental Markov systems,, , ().   Google Scholar

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