TOPOLOGICAL-ENTROPY OF STANDARD TYPE MONOTONE TWIST MAPS

We study invariant measures of families of monotone twist maps S-gamma (g,p) = (2q-gamma . V-1(q),q) with periodic Morse potential V. We prov e that there exist a constant C = C(V) such that the topological entro py satisfies h(top)(Sr) greater than or equal to log(C . gamma)/3 In p articular, h(top)(S-gamma) --> infinity for \y\ --> infinity. We show also that there exist arbitrary large gamma such that S-gamma has nonu niformly hyperbolic invariant measures mu(gamma) with positive metric entropy. For large gamma, the measures mu(gamma) are hyperbolic and, f or a class of potentials which includes V(q) = sin(q), the Lyapunov ex ponent of the map S with invariant measure mu(gamma) grows monotonical ly with gamma.