From Anosov Closing Lemma to global data of cohomological nature

Abstract

For diffeomorphisms with hyperbolic sets, the Anosov Closing Lemma ensures the existence of periodic orbits in the neighbourhood of orbits that return close enough to themselves. Moreover, it defines how is controlled the distance between corresponding points of an initial orbit and the constructed periodic orbits. In the essential, this article presents proof of the estimate of this distance. The Anosov Closing Lemma is crucial in the statement of Livschitz Theorem that, based only on the periodic data, provides a necessary and sufficient condition so that cohomological equations have sufficiently regular solutions, Hölder solutions. It is one of the main tools to obtain global data of cohomological nature based only on periodic data. As suggested by Katok and Hasselblat in \cite{KH}, it is demonstrated, in detail and in the cohomology context, the Livschitz Theorem for hyperbolic diffeomorphisms, where the mentioned distance control inequality is essential.