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

DOI:
https://doi.org/10.1090/S0002-9939-96-03357-6

MathSciNet review:
1327031

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 in a certain class of piecewise linear Bernoulli toral linked twist maps, given any there is a Bernoulli toral linked twist map with positive Lyapunov exponents defined only on a set of measure zero such that is within of in the metric.

**1.**N. Chernov and C. Haskell,*Nonuniformly hyperbolic systems are Bernoulli*, Ergodic Theory and Dynamical Systems (to appear).**2.**G. Gallavotti and D.S Ornstein,*Billiards and Bernoulli schemes*, Comm. Math. Phys.**38**(1974), 83--101. MR**50:7480****3.**A. Katok, J.M. Strelcyn, F. Ledrappier, and F. Przytycki,*Invariant manifolds, entropy and billiards, smooth maps with singularities*, Lecture Notes in Math., vol. 1222, Springer-Verlag, 1986. MR**88k:58075****4.**C. Liverani and M. Wojtkowski,*Ergodicity in Hamiltonian systems*(preprint).**5.**M. Nicol,*Stochastic stability of Bernoulli toral linked twist maps of finite and infinite entropy*, Ergodic Theory and Dynamical Systems, (to appear).**6.**D. S. Ornstein and B. Weiss. Statistical properties of chaotic systems.*Bull. Amer. Math. Soc.*(N.S.) 24, 11-116, 1991. MR**91g:58160****7.**D. S. Ornstein and B. Weiss. Geodesic flows are Bernoullian.*Israel J. Math*14, 184-198, 1973. MR**48:4272****8.**F. Przytycki. Ergodicity of total linked twist mappings.*Ann. Scient. Ec. Norm. Sup.*(4) 16, 345-355, 1983. MR**85k:58051****9.**P. Walters.*An introduction to ergodic theory*, Graduate Texts in Mathematics, Volume 79, Springer Verlag, 1982. MR**84e:28017****10.**M. P. Wojtkowski,*Invariant families of cones and Lyapunov exponents*, Erg. Th. Dyn. Syst.**5**(1985), 145--161. MR**86h:58090**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
58F11,
58F15

Retrieve articles in all journals with MSC (1991): 58F11, 58F15

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:
https://doi.org/10.1090/S0002-9939-96-03357-6

Keywords:
Lyapunov exponent,
linked twist map

Received by editor(s):
June 2, 1994

Communicated by:
Linda Keen

Article copyright:
© Copyright 1996
American Mathematical Society