95-192 Knill O.

Topological entropy of Standard type monotone twist maps (51K, LaTeX) Apr 11, 95

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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 exists a constant $C=C(V)$ such that the topologlical 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 exists arbitrary large $\gamma$ such that $S_{\gamma}$ has nonuniformly hyperbolic invariant measures $\mu_{\gamma}$ with positive metric entropy. For larger $\gamma$, the measures $\mu_{\gamma}$ become hyperbolic and the Lyapunov exponent of the map $S$ with invariant measure $\mu_{\gamma}$ grows monotonically with $\gamma$.

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