Math Seminars
J. Buzzi
(Ecole Polytechnique)
Hyperbolicity from Entropies: some results through inducing schemes
Entropy assumptions ensure some kind of hyperbolicity in low dimension as
examplified by the Ruelle-Margulis inequality and Katok's (approximation)
theorem on surface diffeomorphisms. For piecewise interval maps, Hofbauer
obtained an even stronger (isomorphism) theorem, yielding, e.g., control
of the measures of maximum entropy. We previously showed how this strategy
could be extended to high dimensional smooth maps assuming only a robust
entropy "entropy-expansion" condition.
We shall discuss some hyperbolic versions of this approach, both in
dimension 2 (arbitrary piecewise affine surface homeomorphisms) and in
high dimensions (under an assumption of dominated splitting). We will
explain how entropy bounds and inducing give a way around the (explict)
building of Markov partitions.