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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Bernoulli toral linked twist map
without positive Lyapunov exponents


Author: Matthew Nicol
Journal: Proc. Amer. Math. Soc. 124 (1996), 1253-1263
MSC (1991): Primary 58F11; Secondary 58F15
MathSciNet review: 1327031
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Abstract: The presence of positive Lyapunov exponents in a dynamical system is often taken to be equivalent to the chaotic behavior of that system. We construct a Bernoulli toral linked twist map which has positive Lyapunov exponents and local stable and unstable manifolds defined only on a set of measure zero. This is a deterministic dynamical system with the strongest stochastic property, yet it has positive Lyapunov exponents only on a set of measure zero. In fact we show that for any map $g$ in a certain class of piecewise linear Bernoulli toral linked twist maps, given any $\epsilon >0$ there is a Bernoulli toral linked twist map $g'$ with positive Lyapunov exponents defined only on a set of measure zero such that $g'$ is within $\epsilon$ of $g$ in the $\bar{d}$ metric.


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Additional Information

Matthew Nicol
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3476 \indent{E-mail address}: nicol@math.uh.edu
Address at time of publication: Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, United Kingdom
Email: nicol@math.uh.edu, matt@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03357-6
PII: S 0002-9939(96)03357-6
Keywords: Lyapunov exponent, linked twist map
Received by editor(s): June 2, 1994
Communicated by: Linda Keen
Article copyright: © Copyright 1996 American Mathematical Society