Anatole Katok, Dissertation Advisor/Co-Advisor Anatole Katok, Committee Chair/Co-Chair Yakov B Pesin, Committee Member Omri Sarig, Committee Member Asok Ray, Committee Member
Keywords:
entropy measure skew product Lyapunove exponent
Abstract:
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 cite{Ka80} 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. We can estimate the entropy of the induced measures and show that they produce ergodic measures with 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.
