Communications in Mathematical Physics

, Volume 359, Issue 1, pp 347–373 | Cite as

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

  • Alex Blumenthal
  • Jinxin Xue
  • Lai-Sang Young


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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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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