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

Random graph directed Markov systems

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.
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]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[2]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[3]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399

[4]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[5]

The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013

[6]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[7]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004

[8]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

[9]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[10]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[11]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[12]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151

[13]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[14]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128

[15]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283

[16]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018

[17]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[18]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[19]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[20]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]