September  2011, 31(3): 669-684. doi: 10.3934/dcds.2011.31.669

Extensive escape rate in lattices of weakly coupled expanding maps

1. 

Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS - Université de Rouen, Avenue de l’Université, 76801 Saint Étienne du Rouvray, France

2. 

Centre de Physique Théorique, UMR 6207 CNRS - Université Aix-Marseille II, Campus de Luminy Case 907, 13288 Marseille CEDEX 9, France

Received  February 2011 Revised  May 2011 Published  August 2011

In this paper, we study the escape rate of infinite lattices of weakly coupled maps with uniformly expanding repeller. In particular, it is proved that the escape rate of spatially periodic approximations is extensive and grows linearly with the period size. The proof relies on symbolic dynamics and is based on the control of cumulative effects of perturbations in cylinder sets with distinct spatial periods. A piecewise affine diffusive example is presented that exhibits monotonic decay of the escape rate with coupling intensity.
Citation: Jean-Baptiste Bardet, Bastien Fernandez. Extensive escape rate in lattices of weakly coupled expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 669-684. doi: 10.3934/dcds.2011.31.669
References:
[1]

V. Afraimovich, M. Courbage, B. Fernandez and A. Morante, Directional entropy in lattice dynamical systems,, in, (2002), 9.   Google Scholar

[2]

V. Afraimovich and B. Fernandez, Topological properties of linearly coupled expanding map lattices,, Nonlinearity, 13 (2000), 973.  doi: 10.1088/0951-7715/13/4/301.  Google Scholar

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V. Baladi, C. Bonatti and B. Schmitt, Abnormal escape rates from nonuniformly hyperbolic sets,, Ergod. Th. & Dynam. Sys., 19 (1999), 1111.   Google Scholar

[6]

H. Bruin, M. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes,, Ergod. Th. & Dynam. Sys., 30 (2010), 687.   Google Scholar

[7]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181.  doi: 10.1007/BF01389848.  Google Scholar

[8]

L. Bunimovich and Y. Sinaĭ, Spacetime chaos in coupled map lattices,, Nonlinearity, 1 (1988), 491.  doi: 10.1088/0951-7715/1/4/001.  Google Scholar

[9]

J.-R. Chazottes and B. Fernandez, eds., "Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems,", Lect. Notes Phys., 671 (2005).   Google Scholar

[10]

N. Chernov, R. Markarian and S. Troubetzkoy, Conditionally invariant measures for Anosov maps with small holes,, Ergod. Th. & Dynam. Sys., 18 (1998), 1049.   Google Scholar

[11]

P. Collet, Some ergodic properties of maps of the interval,, in, 52 (1991), 55.   Google Scholar

[12]

P. Collet and J.-P. Eckmann, The definition and measurment of the topological entropy per unit volume in parabolic PDEs,, Nonlinearity, 12 (1999), 451.  doi: 10.1088/0951-7715/12/3/002.  Google Scholar

[13]

P. Collet, S. Martinez and B. Schmitt, The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems,, Nonlinearity, 7 (1994), 1437.  doi: 10.1088/0951-7715/7/5/010.  Google Scholar

[14]

M. Demers and L.-S. Young, Escape rates and conditionally invariant measures,, Nonlinearity, 19 (2006), 377.  doi: 10.1088/0951-7715/19/2/008.  Google Scholar

[15]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Rev. Mod. Phys., 57 (1985), 617.  doi: 10.1103/RevModPhys.57.617.  Google Scholar

[16]

B. Fernandez and P. Guiraud, Route to chaotic synchronisation in coupled map lattices: Rigorous results,, Discrete Continuous Dynam. Systems Ser. B, 4 (2004), 435.  doi: 10.3934/dcdsb.2004.4.435.  Google Scholar

[17]

P. Gaspard and J. R. Dorfman, Chaotic scattering theory, thermodynamic formalism, and transport coefficients,, Phys. Rev. E, 52 (1995), 3525.  doi: 10.1103/PhysRevE.52.3525.  Google Scholar

[18]

K. Kaneko, ed., "Theory and Applications of Coupled Map Lattices,", Nonlinear Science: Theory and Applications, (1993).   Google Scholar

[19]

G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension,, Commun. Math. Phys., 262 (2006), 33.  doi: 10.1007/s00220-005-1474-7.  Google Scholar

[20]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[21]

Y. Pesin and Y. Sinaĭ, "Space-Time Chaos in Chains of Weakly Interacting Hyperbolic Mappings,", Adv. Soviet Math., 3 (1991), 165.   Google Scholar

[22]

G. Pianigiani and J. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos,, Trans. Amer. Math. Soc., 252 (1979), 351.   Google Scholar

show all references

References:
[1]

V. Afraimovich, M. Courbage, B. Fernandez and A. Morante, Directional entropy in lattice dynamical systems,, in, (2002), 9.   Google Scholar

[2]

V. Afraimovich and B. Fernandez, Topological properties of linearly coupled expanding map lattices,, Nonlinearity, 13 (2000), 973.  doi: 10.1088/0951-7715/13/4/301.  Google Scholar

[3]

V. Afraimovich, A. Morante and E. Ugalde, On the density of directional entropy in lattice dynamical systems,, Nonlinearity, 17 (2004), 105.  doi: 10.1088/0951-7715/17/1/007.  Google Scholar

[4]

C. Baesens and R. MacKay, Exponential localization of linear response in networks with exponentially decaying coupling,, Nonlinearity, 10 (1997), 931.  doi: 10.1088/0951-7715/10/4/008.  Google Scholar

[5]

V. Baladi, C. Bonatti and B. Schmitt, Abnormal escape rates from nonuniformly hyperbolic sets,, Ergod. Th. & Dynam. Sys., 19 (1999), 1111.   Google Scholar

[6]

H. Bruin, M. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes,, Ergod. Th. & Dynam. Sys., 30 (2010), 687.   Google Scholar

[7]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181.  doi: 10.1007/BF01389848.  Google Scholar

[8]

L. Bunimovich and Y. Sinaĭ, Spacetime chaos in coupled map lattices,, Nonlinearity, 1 (1988), 491.  doi: 10.1088/0951-7715/1/4/001.  Google Scholar

[9]

J.-R. Chazottes and B. Fernandez, eds., "Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems,", Lect. Notes Phys., 671 (2005).   Google Scholar

[10]

N. Chernov, R. Markarian and S. Troubetzkoy, Conditionally invariant measures for Anosov maps with small holes,, Ergod. Th. & Dynam. Sys., 18 (1998), 1049.   Google Scholar

[11]

P. Collet, Some ergodic properties of maps of the interval,, in, 52 (1991), 55.   Google Scholar

[12]

P. Collet and J.-P. Eckmann, The definition and measurment of the topological entropy per unit volume in parabolic PDEs,, Nonlinearity, 12 (1999), 451.  doi: 10.1088/0951-7715/12/3/002.  Google Scholar

[13]

P. Collet, S. Martinez and B. Schmitt, The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems,, Nonlinearity, 7 (1994), 1437.  doi: 10.1088/0951-7715/7/5/010.  Google Scholar

[14]

M. Demers and L.-S. Young, Escape rates and conditionally invariant measures,, Nonlinearity, 19 (2006), 377.  doi: 10.1088/0951-7715/19/2/008.  Google Scholar

[15]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Rev. Mod. Phys., 57 (1985), 617.  doi: 10.1103/RevModPhys.57.617.  Google Scholar

[16]

B. Fernandez and P. Guiraud, Route to chaotic synchronisation in coupled map lattices: Rigorous results,, Discrete Continuous Dynam. Systems Ser. B, 4 (2004), 435.  doi: 10.3934/dcdsb.2004.4.435.  Google Scholar

[17]

P. Gaspard and J. R. Dorfman, Chaotic scattering theory, thermodynamic formalism, and transport coefficients,, Phys. Rev. E, 52 (1995), 3525.  doi: 10.1103/PhysRevE.52.3525.  Google Scholar

[18]

K. Kaneko, ed., "Theory and Applications of Coupled Map Lattices,", Nonlinear Science: Theory and Applications, (1993).   Google Scholar

[19]

G. Keller and C. Liverani, Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension,, Commun. Math. Phys., 262 (2006), 33.  doi: 10.1007/s00220-005-1474-7.  Google Scholar

[20]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[21]

Y. Pesin and Y. Sinaĭ, "Space-Time Chaos in Chains of Weakly Interacting Hyperbolic Mappings,", Adv. Soviet Math., 3 (1991), 165.   Google Scholar

[22]

G. Pianigiani and J. Yorke, Expanding maps on sets which are almost invariant: Decay and chaos,, Trans. Amer. Math. Soc., 252 (1979), 351.   Google Scholar

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