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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 and Continuous Dynamical Systems, 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, Games and Science II, Springer Proceedings in Mathematics, 2 (2011), 161-185. doi: 10.1007/978-3-642-14788-3_13.

[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), 367-394.

[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), 589-590.

[4]

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

[5]

M. Denker and M. Urbański, On the existence of conformal measures, Trans. Am. Math. Soc., 328 (1991), 563-587. doi: 10.1090/S0002-9947-1991-1014246-4.

[6]

H. Föllmer, U. Horst and A. Kirman, Equilibria in financial markets with heterogeneous agents: A probabilistic perspective, Journal of Mathematical Economics, 41 (2005), 123-155. doi: 10.1016/j.jmateco.2004.08.001.

[7]

K. Horbacz and T. Szarek, Irreducible Markov systems on Polish spaces, Studia Math., 177 (2006), 285-295. doi: 10.4064/sm177-3-7.

[8]

R. Isaac, Markov processes and unique stationary probability measures, Pacific J. Math., 12 (1962), 273-286. doi: 10.2140/pjm.1962.12.273.

[9]

A. Johansson, A. Öberg and M. Pollicott, Countable state shifts and uniqueness of g-measures, Amer. J. Math., 129 (2007), 1501-1511. doi: 10.1353/ajm.2007.0044.

[10]

M. Keane, Strongly Mixing $g$-Measures, Inventiones math., 16 (1972), 309-324. doi: 10.1007/BF01425715.

[11]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 30 (1974), 185-202. doi: 10.1007/BF00533471.

[12]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Th. & Dynam. Sys., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[13]

O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel Journal of Mathematics, 121 (2001), 285-311. doi: 10.1007/BF02802508.

[14]

W. Slomczynski, Dynamical Entropy, Markov Operators, and Iterated Function Systems, Rozprawy Habilitacyjne Uniwersytetu Jagiellońskiego Nr 362, Wydawnictwo Uniwersytetu Jagiellońskiego, Kraków, 2003.

[15]

T. Szarek, Invariant measures for nonexpansive Markov operators on Polish spaces, Diss. Math., 415 (2003), 1-62. doi: 10.4064/dm415-0-1.

[16]

P. Walters, Ruelle's Operator Theorem and $g$-measures, Tran. AMS, 214 (1975), 375-387. doi: 10.1090/S0002-9947-1975-0412389-8.

[17]

I. Werner, Contractive Markov systems, J. London Math. Soc., 71 (2005), 236-258. doi: 10.1112/S0024610704006088.

[18]

I. Werner, Coding map for a contractive Markov system, Math. Proc. Camb. Phil. Soc., 140 (2006), 333-347. doi: 10.1017/S0305004105009072.

[19]

I. Werner, The generalized Markov measure as an equilibrium state, Nonlinearity 18 (2005), 2261-2274. doi: 10.1088/0951-7715/18/5/019.

[20]

I. Werner, Dynamically defined measures and equilibrium states, J. Math. Phys. 52 (2011), 122701, 17 pp. doi: 10.1063/1.3666020.

[21]

I. Werner, Erratum: Dynamically defined measures and equilibrium states, J. Math. Phys. 53 (2012), 079902, 2 pp. doi: 10.1063/1.4736999.

[22]

I. Werner, Contractive Markov systems II, arXiv:math/0503633.

[23]

I. Werner, Fundamental Markov systems, arXiv:math/0509120.

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, Games and Science II, Springer Proceedings in Mathematics, 2 (2011), 161-185. doi: 10.1007/978-3-642-14788-3_13.

[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), 367-394.

[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), 589-590.

[4]

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

[5]

M. Denker and M. Urbański, On the existence of conformal measures, Trans. Am. Math. Soc., 328 (1991), 563-587. doi: 10.1090/S0002-9947-1991-1014246-4.

[6]

H. Föllmer, U. Horst and A. Kirman, Equilibria in financial markets with heterogeneous agents: A probabilistic perspective, Journal of Mathematical Economics, 41 (2005), 123-155. doi: 10.1016/j.jmateco.2004.08.001.

[7]

K. Horbacz and T. Szarek, Irreducible Markov systems on Polish spaces, Studia Math., 177 (2006), 285-295. doi: 10.4064/sm177-3-7.

[8]

R. Isaac, Markov processes and unique stationary probability measures, Pacific J. Math., 12 (1962), 273-286. doi: 10.2140/pjm.1962.12.273.

[9]

A. Johansson, A. Öberg and M. Pollicott, Countable state shifts and uniqueness of g-measures, Amer. J. Math., 129 (2007), 1501-1511. doi: 10.1353/ajm.2007.0044.

[10]

M. Keane, Strongly Mixing $g$-Measures, Inventiones math., 16 (1972), 309-324. doi: 10.1007/BF01425715.

[11]

F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 30 (1974), 185-202. doi: 10.1007/BF00533471.

[12]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Th. & Dynam. Sys., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[13]

O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel Journal of Mathematics, 121 (2001), 285-311. doi: 10.1007/BF02802508.

[14]

W. Slomczynski, Dynamical Entropy, Markov Operators, and Iterated Function Systems, Rozprawy Habilitacyjne Uniwersytetu Jagiellońskiego Nr 362, Wydawnictwo Uniwersytetu Jagiellońskiego, Kraków, 2003.

[15]

T. Szarek, Invariant measures for nonexpansive Markov operators on Polish spaces, Diss. Math., 415 (2003), 1-62. doi: 10.4064/dm415-0-1.

[16]

P. Walters, Ruelle's Operator Theorem and $g$-measures, Tran. AMS, 214 (1975), 375-387. doi: 10.1090/S0002-9947-1975-0412389-8.

[17]

I. Werner, Contractive Markov systems, J. London Math. Soc., 71 (2005), 236-258. doi: 10.1112/S0024610704006088.

[18]

I. Werner, Coding map for a contractive Markov system, Math. Proc. Camb. Phil. Soc., 140 (2006), 333-347. doi: 10.1017/S0305004105009072.

[19]

I. Werner, The generalized Markov measure as an equilibrium state, Nonlinearity 18 (2005), 2261-2274. doi: 10.1088/0951-7715/18/5/019.

[20]

I. Werner, Dynamically defined measures and equilibrium states, J. Math. Phys. 52 (2011), 122701, 17 pp. doi: 10.1063/1.3666020.

[21]

I. Werner, Erratum: Dynamically defined measures and equilibrium states, J. Math. Phys. 53 (2012), 079902, 2 pp. doi: 10.1063/1.4736999.

[22]

I. Werner, Contractive Markov systems II, arXiv:math/0503633.

[23]

I. Werner, Fundamental Markov systems, arXiv:math/0509120.

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