American Institute of Mathematical Sciences

Advanced
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,, , (). 

[23]

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

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,, , (). 

[23]

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

[1]

Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261
[2]

Manfred G. Madritsch. Non-normal numbers with respect to Markov partitions. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 663-676. doi: 10.3934/dcds.2014.34.663
[3]

Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018
[4]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627
[5]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111
[6]

Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003
[7]

Thomas Ward, Yuki Yayama. Markov partitions reflecting the geometry of $\times2$, $\times3$. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 613-624. doi: 10.3934/dcds.2009.24.613
[8]

Yanqing Liu, Yanyan Yin, Kok Lay Teo, Song Wang, Fei Liu. Probabilistic control of Markov jump systems by scenario optimization approach. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1447-1453. doi: 10.3934/jimo.2018103
[9]

Fernando J. Sánchez-Salas. Dimension of Markov towers for non uniformly expanding one-dimensional systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1447-1464. doi: 10.3934/dcds.2003.9.1447
[10]

Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial and Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124
[11]

Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533
[12]

Johnathan M. Bardsley. Gaussian Markov random field priors for inverse problems. Inverse Problems and Imaging, 2013, 7 (2) : 397-416. doi: 10.3934/ipi.2013.7.397
[13]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131
[14]

Felix X.-F. Ye, Yue Wang, Hong Qian. Stochastic dynamics: Markov chains and random transformations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2337-2361. doi: 10.3934/dcdsb.2016050
[15]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[16]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[17]

Peter E. Kloeden, Victor Kozyakin. Asymptotic behaviour of random tridiagonal Markov chains in biological applications. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 453-465. doi: 10.3934/dcdsb.2013.18.453
[18]

Tom Goldstein, Xavier Bresson, Stan Osher. Global minimization of Markov random fields with applications to optical flow. Inverse Problems and Imaging, 2012, 6 (4) : 623-644. doi: 10.3934/ipi.2012.6.623
[19]

Hisayoshi Toyokawa. $\sigma$-finite invariant densities for eventually conservative Markov operators. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2641-2669. doi: 10.3934/dcds.2020144
[20]

Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (115)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy