| We study entropy and invariant measures for smooth diffeomorphisms. The main result of the dissertation establishes a theorem on skew product maps with diffeomorphisms on fibers. We show if, for an ergodic invariant measure mu, all Lyapunov exponents along the fibers are non-zero, then any value, between 0 and the metric entropy of mu, is the metric entropy of an ergodic invariant measure for the map. This result generalizes a famous result of A. Katok [17] in which mu is required to be a hyperbolic measure.;To construct the measures of intermediate entropies we find an invariant set on which the induced map is also a skew product which acts like horseshoe maps on fibers. From this set we can construct ergodic measures with the maximal entropy arbitrarily close to the entropy of mu. Since a horseshoe map is conjugate to a full shift, all intermediate entropies can be obtained by changing the weights of different symbols in a continuous way. |
