# American Institute of Mathematical Sciences

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

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##### 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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