American Institute of Mathematical Sciences

Advanced
April  2011, 30(1): 261-298. doi: 10.3934/dcds.2011.30.261

Random graph directed Markov systems

Mario Roy 1, and  Mariusz Urbański 2,

1. 

Glendon College, York University, 2275 Bayview Avenue, Toronto
2. 

Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received  January 2010 Revised  October 2010 Published  February 2011

We introduce and explore random conformal graph directed Mar-kov systems governed by measure-preserving ergodic dynamical systems. We first develop the symbolic thermodynamic formalism for random finitely primitive subshifts of finite type with a countable alphabet (by establishing tightness in a narrow topology). We then construct fibrewise conformal and invariant measures along with fibrewise topological pressure. This enables us to define the expected topological pressure $\mathcal EP(t)$ and to prove a variant of Bowen's formula which identifies the Hausdorff dimension of almost every limit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is the unique zero of the expected pressure if the alphabet is finite or the system is regular. We introduce the class of essentially random systems and we show that in the realm of systems with finite alphabet their limit set fibers are never homeomorphic in a bi-Lipschitz fashion to the limit sets of deterministic systems; they thus make up a drastically new world. We also provide a large variety of examples, with exact computations of Hausdorff dimensions, and we study in detail the small random perturbations of an arbitrary elliptic function.
Keywords: random dynamical systems, elliptic functions., thermodynamic formalism, Iterated function systems, limit set, countable shifts, Hausdorff dimension.
Mathematics Subject Classification: Primary: 37H99, 37D35, 37F35; Secondary: 37F1.
Citation: Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261
References:
[1]

A. Berlinkov and R. D. Mauldin, Packing measure and dimension of random fractals,, J. Theoret. Probab., 15 (2002), 695.  doi: 10.1023/A:1016271916074.  Google Scholar

[2]

T. Bogenschuetz and V. M. Gundlach, Ruelle's transfer operator for random subshifts of finite type,, Ergod. Th. & Dynam. Sys., 15 (1995), 413.   Google Scholar

[3]

H. Crauel, "Random Probability Measures on Polish Spaces,", Stochastics Monographs, 11 (2002).   Google Scholar

[4]

M. Denker, Y. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts,, Discrete Contin. Dyn. Syst., 22 (2008), 131.  doi: 10.3934/dcds.2008.22.131.  Google Scholar

[5]

S. Graf, Statistically self-similar fractals,, Probab. Theory Related Fields, 74 (1987), 357.  doi: 10.1007/BF00699096.  Google Scholar

[6]

J. Kotus and M. Urbański, Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions,, Bull. London Math. Soc., 35 (2003), 269.  doi: 10.1112/S0024609302001686.  Google Scholar

[7]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[8]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Tracts in Mathematics, 148 (2003).   Google Scholar

[9]

R. D. Mauldin and S. C. Williams, Random recursive constructions: Asymptotic geometric and topological properties,, Trans. Amer. Math. Soc., 295 (1986), 325.  doi: 10.1090/S0002-9947-1986-0831202-5.  Google Scholar

[10]

V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry,, preprint, (2008).   Google Scholar

[11]

M. Stadlbauer, On random topological Markov chains with big images and preimages,, Stoch. Dyn., 10 (2010), 77.  doi: 10.1142/S0219493710002863.  Google Scholar

[12]

B. Stratmann and M. Urbański, Pseudo-Markov systems and infinitely generated Schottky groups,, Amer. J. Math., 129 (2007), 1019.  doi: 10.1353/ajm.2007.0028.  Google Scholar

show all references

References:
[1]

A. Berlinkov and R. D. Mauldin, Packing measure and dimension of random fractals,, J. Theoret. Probab., 15 (2002), 695.  doi: 10.1023/A:1016271916074.  Google Scholar

[2]

T. Bogenschuetz and V. M. Gundlach, Ruelle's transfer operator for random subshifts of finite type,, Ergod. Th. & Dynam. Sys., 15 (1995), 413.   Google Scholar

[3]

H. Crauel, "Random Probability Measures on Polish Spaces,", Stochastics Monographs, 11 (2002).   Google Scholar

[4]

M. Denker, Y. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts,, Discrete Contin. Dyn. Syst., 22 (2008), 131.  doi: 10.3934/dcds.2008.22.131.  Google Scholar

[5]

S. Graf, Statistically self-similar fractals,, Probab. Theory Related Fields, 74 (1987), 357.  doi: 10.1007/BF00699096.  Google Scholar

[6]

J. Kotus and M. Urbański, Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions,, Bull. London Math. Soc., 35 (2003), 269.  doi: 10.1112/S0024609302001686.  Google Scholar

[7]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[8]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Tracts in Mathematics, 148 (2003).   Google Scholar

[9]

R. D. Mauldin and S. C. Williams, Random recursive constructions: Asymptotic geometric and topological properties,, Trans. Amer. Math. Soc., 295 (1986), 325.  doi: 10.1090/S0002-9947-1986-0831202-5.  Google Scholar

[10]

V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry,, preprint, (2008).   Google Scholar

[11]

M. Stadlbauer, On random topological Markov chains with big images and preimages,, Stoch. Dyn., 10 (2010), 77.  doi: 10.1142/S0219493710002863.  Google Scholar

[12]

B. Stratmann and M. Urbański, Pseudo-Markov systems and infinitely generated Schottky groups,, Amer. J. Math., 129 (2007), 1019.  doi: 10.1353/ajm.2007.0028.  Google Scholar

[1]

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

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[3]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
[4]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
[5]

Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303
[6]

Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1
[7]

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
[8]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475
[9]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[10]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[11]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[12]

Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647
[13]

Pablo G. Barrientos, Abbas Fakhari, Aliasghar Sarizadeh. Density of fiberwise orbits in minimal iterated function systems on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3341-3352. doi: 10.3934/dcds.2014.34.3341
[14]

Vaughn Climenhaga. Multifractal formalism derived from thermodynamics for general dynamical systems. Electronic Research Announcements, 2010, 17: 1-11. doi: 10.3934/era.2010.17.1
[15]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829
[16]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[17]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
[18]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[19]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[20]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences