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Stochastic stability of Bernoulli toral linked twist maps of finite and infinite entropy

Published online by Cambridge University Press:  19 September 2008

Matthew Nicol
Matthew Nicol
Affiliation:
Department of Mathematics, UMIST, PO Box 88, Sackville Street, Manchester M60 1QD, UK (e-mail: M.Nicol@umist.ac.uk)
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Abstract

We construct linked twist maps of the two-dimensional torus which are Bernoulli and possess infinite entropy. Inparticular, we construct a Bernoulli toral linked twist map B* of infinite entropy which has smooth, absolutely continuous local (un)stable manifolds and positive Lyapunov exponents defined almost everywhere. This map is continuous at each point save those on two line segments.

B* is shown to be stochastically stable under the following random perturbation: apply the map to a point p and then jump (all points move the same distance and in the same direction) according to a B-process (not necessarily an independent process) such that the expected distance moved is equal to r. Stochastic stability means that given α > 0 if r > 0 is sufficiently small then the perturbed and unperturbed systems are α-congruent. We prove a similar stability result for B* under a perturbation in which the random jump described above is distributed according to a general stochastic process. These stability results are also shown to hold (in a slightly modified form) for a general class of finite-entropy toral linked twist maps under the same perturbations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 3 , June 1996 , pp. 493 - 518
Copyright
Copyright © Cambridge University Press 1996

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