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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Topological entropy of standard type monotone twist maps
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by Oliver Knill PDF
Trans. Amer. Math. Soc. 348 (1996), 2999-3013 Request permission


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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Additional Information
  • Oliver Knill
  • Affiliation: Division of Physics, Mathematics and Astronomy, California Institute of Technology, 91125 Pasadena, California
  • Email:
  • 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.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 2999-3013
  • MSC (1991): Primary 58F11, 28D20, 28D10; Secondary 58E30, 58F05
  • DOI:
  • MathSciNet review: 1373642