Topological Entropy of Standard Type Monotone Twist Maps

We study invariant measures of families of monotone twist maps Sγ(q, p) = (2q−p+γ ·V ′(q), q) with periodic Morse potential V . We prove that there exist a constant C = C(V ) such that the topological entropy satisfies htop(Sγ) ≥ log(C · γ)/3. In particular, htop(Sγ) → ∞ for |γ| → ∞. We show also that there exist arbitrary large γ such that Sγ has nonuniformly hyperbolic invariant measures μγ with positive metric entropy. For large γ, the measures μγ are hyperbolic and, for a class of potentials which includes V (q) = sin(q), the Lyapunov exponent of the map S with invariant measure μγ grows monotonically with γ.