Knill O.
Topological entropy of Standard type monotone twist maps
(51K, LaTeX)
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 $\gamma$.