TY - JOUR

T1 - Lyapunov Exponents and Correlation Decay for Random Perturbations of Some Prototypical 2D Maps

AU - Blumenthal, Alex

AU - Xue, Jinxin

AU - Young, Lai Sang

N1 - Funding Information:
Alex Blumenthal: This research was supported by NSF Grant DMS-1604805. Jinxin Xue: This research was supported by NSF Grant DMS-1500897. Lai-Sang Young: This research was supported in part by NSF Grant DMS-1363161.
Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large—but non-invariant—subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth.

AB - To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large—but non-invariant—subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth.

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U2 - 10.1007/s00220-017-2999-2

DO - 10.1007/s00220-017-2999-2

M3 - Article

AN - SCOPUS:85031410240

VL - 359

SP - 347

EP - 373

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -