Topological entropy of standard type monotone twist maps

Knill, Oliver

doi:10.1090/S0002-9947-96-01728-X

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

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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