Hostname: page-component-77f85d65b8-45ctf Total loading time: 0 Render date: 2026-04-20T09:18:08.233Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Abramov, L. and Rohlin, V. A.. The entropy of a skew product of measure-preserving transformations. Vestnik Leningrad Univ. 17 (1962), 5–13.Google Scholar
[2]Burton, R. and Easton, R.. Ergodicity of linked twist maps. Global Theory of Dynamical Systems 819 (1980), 35–49.Google Scholar
[3]Chernov, N. and Haskell, C.. Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Th. & Dynam. Sys. 16 (1996), 19‐44.CrossRefGoogle Scholar
[4]Katok, A., Strelcyn, J. M., Ledrappier, F. and Przytycki, F.. Invariant manifolds, entropy and billiards, smooth maps with singularities (Lecture Notes in Mathematics 1222). Springer, Berlin, 1986.CrossRefGoogle Scholar
[5]Nicol, M.. A Bernoulli toral linked twist map without positive Lyapunov exponents. Proc. AMS 124 (1996), 1253–1263.(to appear).CrossRefGoogle Scholar
[6]Ornstein, D. S. and Weiss, B.. Statistical properties of chaotic systems. Bull. Am. Math. Soc 24 (1991), 11–116.CrossRefGoogle Scholar
[7]Ornstein, D. S. and Weiss, B.. Geodesic flows are Bemoullian. lsrael J. Math. 14 (1973), 184–198.CrossRefGoogle Scholar
[8]Przytycki, F.. Ergodicity of toral linked twist mappings. Ann. Scient. Ec. Norm. Sup. Series 1 16 (1983), 345–355.Google Scholar
[9]Rudolph, D.. Fundamentals of Measurable Dynamics. Clarendon, Oxford, 1990.Google Scholar
[10]Thouvenot, J. P.. Quelques propriétés des systèmes dynamiques qui se décomposent en un product de deux systèmes dont l'un est un schéma de Bernoulli. Israel J. Math. 21 (1975), 177–207.CrossRefGoogle Scholar
[11]Troubetzkoy, S.. A comparison of elastic and inelastic billiards. Nonlinearity 3 (1990), 947–961.CrossRefGoogle Scholar