Subadditivity of entropies of commuting diffeomorphisms and examples of non-existence of SBR measures.

Abstract

In the first part of this thesis we consider a pair of commuting diffeomorphisms (f,g) of a compact manifold preserving a common Borel probability measure μ. Oseledec's multiplicative ergodic theorem and certain aspects of Pesin's theory are generalized to this setting. Our main result is the subadditivity of measure-theoretic entropy, i.e. h(μ)(fg) ≤ h(μ) (f) + h(μ)(g). Sufficient conditions are given for equality to hold. The subaddivity of topological entropy is also proved, and a counterexample is given to show that this inequality is not valid if the map fails to be differentiable at even one point. The second part of this thesis addresses the question of existence of Sinai-Bowen-Ruelle measures for systems that are not uniformly hyperbolic. We present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. These examples have fixed points at which the larger eigenvalue is equal to one and the smaller eigenvalue is less than one.