Abstract
It is presented and proved a version of Livschitz Theorem for hyperbolic flows pragmatically oriented to the cohomological context. Previously, it is introduced the concept of cocycle and a natural notion of symmetry for cocycles. It is discussed the fundamental relationship between the existence of solutions of cohomological equations and the behavior of the cocycles along periodic orbits. The generalization of this theorem to a class of suspension flows is also discussed and proved. This generalization allows giving a different proof of the Livschitz Theorem for flows based on the construction of Markov systems for hyperbolic flows.
Highlights
This paper presents a continuous time approach to Livschitz Theorem oriented to the study of cohomology in dynamical systems
This article does not focus on Anosov Closing lemma, it is worth emphasizing that this result is crucial in the statement of the Livschitz Theorem and in ensuring the existence of sufficiently regular solutions of cohomological equations
For flows with hyperbolic sets, this lemma establishes how the distance between corresponding points of an initial orbit and the constructed periodic orbits is controlled. It formalizes how the combination of local hyperbolicity, coming from the linearized dynamical systems analysis, with nontrivial recurrence tends to produces an abundance of periodic orbits
Summary
This paper presents a continuous time approach to Livschitz Theorem oriented to the study of cohomology in dynamical systems. We present cohomological equations in the case of continuous time and discuss the fundamental relationship between the existence of solutions of these equations and the behavior of the cocycles along periodic orbits (Section 2.2). The study of cohomological equations is related in particular to the study of conjugations to an irrational rotation of circle, the existence of absolutely continuous measures for expanding transformations of circle and the topological stability of hyperbolic automorphisms of torus. Such equations arise naturally in celestial mechanics and statistical mechanics. It is one of the main tools for obtaining global cohomological information from periodic information
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