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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, 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), 695-713. 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), 413-447.  Google Scholar

[3]

H. Crauel, "Random Probability Measures on Polish Spaces," Stochastics Monographs, 11, Taylor and Francis, 2002.  Google Scholar

[4]

M. Denker, Y. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 22 (2008), 131-164. doi: 10.3934/dcds.2008.22.131.  Google Scholar

[5]

S. Graf, Statistically self-similar fractals, Probab. Theory Related Fields, 74 (1987), 357-392. 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), 269-275. 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), 105-154. 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, Cambridge, 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), 325-346. doi: 10.1090/S0002-9947-1986-0831202-5.  Google Scholar

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V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry, preprint, 2008, can be found at www.math.unt.edu/ urbanski. Google Scholar

[11]

M. Stadlbauer, On random topological Markov chains with big images and preimages, Stoch. Dyn., 10 (2010), 77-95. 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), 1019-1062. 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), 695-713. 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), 413-447.  Google Scholar

[3]

H. Crauel, "Random Probability Measures on Polish Spaces," Stochastics Monographs, 11, Taylor and Francis, 2002.  Google Scholar

[4]

M. Denker, Y. Kifer and M. Stadlbauer, Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst., 22 (2008), 131-164. doi: 10.3934/dcds.2008.22.131.  Google Scholar

[5]

S. Graf, Statistically self-similar fractals, Probab. Theory Related Fields, 74 (1987), 357-392. 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), 269-275. 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), 105-154. 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, Cambridge, 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), 325-346. 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, can be found at www.math.unt.edu/ urbanski. Google Scholar

[11]

M. Stadlbauer, On random topological Markov chains with big images and preimages, Stoch. Dyn., 10 (2010), 77-95. 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), 1019-1062. doi: 10.1353/ajm.2007.0028.  Google Scholar

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