# 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 and Continuous Dynamical Systems, 2014, 34 (9) : 3747-3759. doi: 10.3934/dcds.2014.34.3747
##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [2] P. H. Baxendale, Stability and equilibrium properties of stochastic flows of diffeomorphisms, in Diffusion Processes and Related Problems in Analysis, Vol. II, Birkhäuser Boston, Boston, MA, 27 (1992), 3-35. [3] G. Birkhoff, Lattice Theory, Third edition, American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. [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), 687-728. doi: 10.1017/S0143385709000200. [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), 353-388. doi: 10.1007/s00220-009-0941-y. [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), 1270-1301. doi: 10.1017/S0143385711000344. [7] M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397. doi: 10.1088/0951-7715/19/2/008. [8] C. Gupta, W. Ott and A. Török, Memory loss for time-dependent piecewise expanding systems in higher dimension, Math. Res. Lett., 20 (2013), 141-161. doi: 10.4310/MRL.2013.v20.n1.a12. [9] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, Vol. 24, Cambridge University Press, Cambridge, 1990. [10] Y. Le Jan, On isotropic Brownian motions, Z. Wahrsch. Verw. Gebiete, 70 (1985), 609-620. [11] F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103. [12] K. K. Lin, E. Shea-Brown and L.-S. Young, Reliability of coupled oscillators, J. Nonlinear Sci., 19 (2009), 497-545. doi: 10.1007/s00332-009-9042-5. [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), 385-412. doi: 10.1016/S0246-0203(02)00005-5. [14] N. Masmoudi and L.-S. Young, Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs, Comm. Math. Phys., 227 (2002), 461-481. doi: 10.1007/s002200200639. [15] J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206 (1999), 273-288. doi: 10.1007/s002200050706. [16] W. Ott, M. Stenlund and L.-S. Young, Memory loss for time-dependent dynamical systems, Math. Res. Lett., 16 (2009), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7. [17] M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity, 24 (2011), 2991-3018. doi: 10.1088/0951-7715/24/10/016. [18] M. Stenlund, L.-S. Young and H.-K. Zhang, Dispersing billiards with moving scatterers, Comm. Math. Phys., 322 (2013), 909-955. doi: 10.1007/s00220-013-1746-6. [19] H. v. d. Bedem and N. Chernov, Expanding maps of an interval with holes, Ergodic Theory Dynam. Systems, 22 (2002), 637-654. doi: 10.1017/S0143385702000329.

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [2] P. H. Baxendale, Stability and equilibrium properties of stochastic flows of diffeomorphisms, in Diffusion Processes and Related Problems in Analysis, Vol. II, Birkhäuser Boston, Boston, MA, 27 (1992), 3-35. [3] G. Birkhoff, Lattice Theory, Third edition, American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. [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), 687-728. doi: 10.1017/S0143385709000200. [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), 353-388. doi: 10.1007/s00220-009-0941-y. [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), 1270-1301. doi: 10.1017/S0143385711000344. [7] M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity, 19 (2006), 377-397. doi: 10.1088/0951-7715/19/2/008. [8] C. Gupta, W. Ott and A. Török, Memory loss for time-dependent piecewise expanding systems in higher dimension, Math. Res. Lett., 20 (2013), 141-161. doi: 10.4310/MRL.2013.v20.n1.a12. [9] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, Vol. 24, Cambridge University Press, Cambridge, 1990. [10] Y. Le Jan, On isotropic Brownian motions, Z. Wahrsch. Verw. Gebiete, 70 (1985), 609-620. [11] F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103. [12] K. K. Lin, E. Shea-Brown and L.-S. Young, Reliability of coupled oscillators, J. Nonlinear Sci., 19 (2009), 497-545. doi: 10.1007/s00332-009-9042-5. [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), 385-412. doi: 10.1016/S0246-0203(02)00005-5. [14] N. Masmoudi and L.-S. Young, Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs, Comm. Math. Phys., 227 (2002), 461-481. doi: 10.1007/s002200200639. [15] J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206 (1999), 273-288. doi: 10.1007/s002200050706. [16] W. Ott, M. Stenlund and L.-S. Young, Memory loss for time-dependent dynamical systems, Math. Res. Lett., 16 (2009), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7. [17] M. Stenlund, Non-stationary compositions of Anosov diffeomorphisms, Nonlinearity, 24 (2011), 2991-3018. doi: 10.1088/0951-7715/24/10/016. [18] M. Stenlund, L.-S. Young and H.-K. Zhang, Dispersing billiards with moving scatterers, Comm. Math. Phys., 322 (2013), 909-955. doi: 10.1007/s00220-013-1746-6. [19] H. v. d. Bedem and N. Chernov, Expanding maps of an interval with holes, Ergodic Theory Dynam. Systems, 22 (2002), 637-654. doi: 10.1017/S0143385702000329.
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