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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topological entropy of standard type monotone twist maps

Author: Oliver Knill
Journal: Trans. Amer. Math. Soc. 348 (1996), 2999-3013
MSC (1991): Primary 58F11, 28D20, 28D10; Secondary 58E30, 58F05
MathSciNet review: 1373642
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Abstract: We study invariant measures of families of monotone twist maps $S_{\gamma }(q,p)$ $=$ $(2q-p+ \gamma \cdot V’(q),q)$ with periodic Morse potential $V$. We prove that there exist a constant $C=C(V)$ such that the topological entropy satisfies $h_{top}(S_{\gamma }) \geq \log (C \cdot \gamma )/3$. In particular, $h_{top}(S_{\gamma }) \to \infty$ for $|\gamma | \to \infty$. We show also that there exist arbitrary large $\gamma$ such that $S_{\gamma }$ has nonuniformly hyperbolic invariant measures $\mu _{\gamma }$ with positive metric entropy. For large $\gamma$, the measures $\mu _{\gamma }$ are hyperbolic and, for a class of potentials which includes $V(q)=\sin (q)$, the Lyapunov exponent of the map $S$ with invariant measure $\mu _{\gamma }$ grows monotonically with $\gamma$.

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Oliver Knill
Affiliation: Division of Physics, Mathematics and Astronomy, California Institute of Technology, 91125 Pasadena, California

Keywords: Ergodic theory, monotone twist maps, topological entropy, hyperbolic sets, Lyapunov exponents, invariant measures, variational principles
Received by editor(s): July 25, 1994
Additional Notes: This material is based upon work which was supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material.
Article copyright: © Copyright 1996 American Mathematical Society