Resumo:
Hyperbolic Dynamics is one of the most important areas in the study of dynamical
systems. Roughly speaking, a hyperbolic set for a dynamic is a non-empty, compact,
and invariant set, such that for every point in this set, the tangent space decomposes
as a direct sum of two subspaces: one stable (uniformly contractive) and one unstable
(uniformly expansive), both of which are invariant under the action of the derivative.
Through the Stable Manifold Theorem, it is known that these subbundles (stable and
unstable) admit local manifolds that are invariant under the dynamics. In the study
of hyperbolic dynamical systems, some classes of dynamics deserve greater emphasis,
namely di eomorphisms of the Axiom A type and Anosov Di eomorphisms. In the case
of Axiom A type di eomorphisms, the Smale Spectral Theorem is valid, a central result in
the study of these types of transformations. Anosov Di eomorphisms and Axiom A type
satisfy shadowing properties, which are indispensable in the study of structural stability in
hyperbolic dynamics. When it comes to structural stability, we can ask about su cient
conditions for the involved topological conjugation to be of class C1. It can be shown
that when an Anosov di eomorphism has the same periodic data as its linearization, then
these two are indeed C1 conjugated, and this can be obtained as an application of Livsic's
Theorem.
