September  2014, 34(9): 3747-3759. doi: 10.3934/dcds.2014.34.3747

Memory loss for nonequilibrium open dynamical systems

1. 

Department of Mathematics, University of Houston, 651 PGH Hall, Houston, TX 77204, United States, United States

Received  March 2013 Revised  December 2013 Published  March 2014

We introduce a notion of conditional memory loss for nonequilibrium open dynamical systems. We prove that this type of memory loss occurs at an exponential rate for nonequilibrium open systems generated by one-dimensional piecewise-differentiable expanding Lasota-Yorke maps. This result may be viewed as a prototype for time-dependent dynamical systems with holes.
Citation: Anushaya Mohapatra, William Ott. Memory loss for nonequilibrium open dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3747-3759. doi: 10.3934/dcds.2014.34.3747
References:
[1]

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

P. H. Baxendale, Stability and equilibrium properties of stochastic flows of diffeomorphisms,, in Diffusion Processes and Related Problems in Analysis, 27 (1992), 3.   Google Scholar

[3]

G. Birkhoff, Lattice Theory,, Third edition, (1967).   Google Scholar

[4]

H. Bruin, M. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes,, Ergodic Theory Dynam. Systems, 30 (2010), 687.  doi: 10.1017/S0143385709000200.  Google Scholar

[5]

M. Demers, P. Wright and L.-S. Young, Escape rates and physically relevant measures for billiards with small holes,, Comm. Math. Phys., 294 (2010), 353.  doi: 10.1007/s00220-009-0941-y.  Google Scholar

[6]

M. F. Demers, P. Wright and L.-S. Young, Entropy, Lyapunov exponents and escape rates in open systems,, Ergodic Theory Dynam. Systems, 32 (2012), 1270.  doi: 10.1017/S0143385711000344.  Google Scholar

[7]

M. F. 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

[8]

C. Gupta, W. Ott and A. Török, Memory loss for time-dependent piecewise expanding systems in higher dimension,, Math. Res. Lett., 20 (2013), 141.  doi: 10.4310/MRL.2013.v20.n1.a12.  Google Scholar

[9]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge Studies in Advanced Mathematics, (1990).   Google Scholar

[10]

Y. Le Jan, On isotropic Brownian motions,, Z. Wahrsch. Verw. Gebiete, 70 (1985), 609.   Google Scholar

[11]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations,, Probab. Theory Related Fields, 80 (1988), 217.  doi: 10.1007/BF00356103.  Google Scholar

[12]

K. K. Lin, E. Shea-Brown and L.-S. Young, Reliability of coupled oscillators,, J. Nonlinear Sci., 19 (2009), 497.  doi: 10.1007/s00332-009-9042-5.  Google Scholar

[13]

C. Liverani and V. Maume-Deschamps, Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set,, Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 385.  doi: 10.1016/S0246-0203(02)00005-5.  Google Scholar

[14]

N. Masmoudi and L.-S. Young, Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs,, Comm. Math. Phys., 227 (2002), 461.  doi: 10.1007/s002200200639.  Google Scholar

[15]

J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity,, Comm. Math. Phys., 206 (1999), 273.  doi: 10.1007/s002200050706.  Google Scholar

[16]

W. Ott, M. Stenlund and L.-S. Young, Memory loss for time-dependent dynamical systems,, Math. Res. Lett., 16 (2009), 463.  doi: 10.4310/MRL.2009.v16.n3.a7.  Google Scholar

[17]

M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms,, Nonlinearity, 24 (2011), 2991.  doi: 10.1088/0951-7715/24/10/016.  Google Scholar

[18]

M. Stenlund, L.-S. Young and H.-K. Zhang, Dispersing billiards with moving scatterers,, Comm. Math. Phys., 322 (2013), 909.  doi: 10.1007/s00220-013-1746-6.  Google Scholar

[19]

H. v. d. Bedem and N. Chernov, Expanding maps of an interval with holes,, Ergodic Theory Dynam. Systems, 22 (2002), 637.  doi: 10.1017/S0143385702000329.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

P. H. Baxendale, Stability and equilibrium properties of stochastic flows of diffeomorphisms,, in Diffusion Processes and Related Problems in Analysis, 27 (1992), 3.   Google Scholar

[3]

G. Birkhoff, Lattice Theory,, Third edition, (1967).   Google Scholar

[4]

H. Bruin, M. Demers and I. Melbourne, Existence and convergence properties of physical measures for certain dynamical systems with holes,, Ergodic Theory Dynam. Systems, 30 (2010), 687.  doi: 10.1017/S0143385709000200.  Google Scholar

[5]

M. Demers, P. Wright and L.-S. Young, Escape rates and physically relevant measures for billiards with small holes,, Comm. Math. Phys., 294 (2010), 353.  doi: 10.1007/s00220-009-0941-y.  Google Scholar

[6]

M. F. Demers, P. Wright and L.-S. Young, Entropy, Lyapunov exponents and escape rates in open systems,, Ergodic Theory Dynam. Systems, 32 (2012), 1270.  doi: 10.1017/S0143385711000344.  Google Scholar

[7]

M. F. 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

[8]

C. Gupta, W. Ott and A. Török, Memory loss for time-dependent piecewise expanding systems in higher dimension,, Math. Res. Lett., 20 (2013), 141.  doi: 10.4310/MRL.2013.v20.n1.a12.  Google Scholar

[9]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge Studies in Advanced Mathematics, (1990).   Google Scholar

[10]

Y. Le Jan, On isotropic Brownian motions,, Z. Wahrsch. Verw. Gebiete, 70 (1985), 609.   Google Scholar

[11]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations,, Probab. Theory Related Fields, 80 (1988), 217.  doi: 10.1007/BF00356103.  Google Scholar

[12]

K. K. Lin, E. Shea-Brown and L.-S. Young, Reliability of coupled oscillators,, J. Nonlinear Sci., 19 (2009), 497.  doi: 10.1007/s00332-009-9042-5.  Google Scholar

[13]

C. Liverani and V. Maume-Deschamps, Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set,, Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 385.  doi: 10.1016/S0246-0203(02)00005-5.  Google Scholar

[14]

N. Masmoudi and L.-S. Young, Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs,, Comm. Math. Phys., 227 (2002), 461.  doi: 10.1007/s002200200639.  Google Scholar

[15]

J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity,, Comm. Math. Phys., 206 (1999), 273.  doi: 10.1007/s002200050706.  Google Scholar

[16]

W. Ott, M. Stenlund and L.-S. Young, Memory loss for time-dependent dynamical systems,, Math. Res. Lett., 16 (2009), 463.  doi: 10.4310/MRL.2009.v16.n3.a7.  Google Scholar

[17]

M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms,, Nonlinearity, 24 (2011), 2991.  doi: 10.1088/0951-7715/24/10/016.  Google Scholar

[18]

M. Stenlund, L.-S. Young and H.-K. Zhang, Dispersing billiards with moving scatterers,, Comm. Math. Phys., 322 (2013), 909.  doi: 10.1007/s00220-013-1746-6.  Google Scholar

[19]

H. v. d. Bedem and N. Chernov, Expanding maps of an interval with holes,, Ergodic Theory Dynam. Systems, 22 (2002), 637.  doi: 10.1017/S0143385702000329.  Google Scholar

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