Shadowing of weakly pseudo-hyperbolic pseudo-orbits in discrete dynamical systems

• Main Author: Gaito, Stephen Thomas
• Published: University of Warwick 1992
• Subjects: 510; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.316479

We consider Cr (r ≥ 1 +γ) diffeomorphisms of compact Riemannian manifolds. Our aim is to develop the analytic machinery required to describe the topological symbolic dynamics of sets of weakly hyperbolic orbits. The Pesin set is an example of such a set. For Axiom-A dynamical systems, that is, for diffeomorphisms which have a uniformly hyperbolic nonwandering set which is the closure of the periodic orbits, this analytic machinery is provided by the Shadowing Lemma. This lemma is a consequence of the Stable Manifold Theorem, and the local product structure of the nonwandering set of an Axiom-A diffeomorphism. Weakly hyperbolic invariant sets, such as the Pesin set, do not, in general, have local product structure. We can however, prove a generalization of the Shadowing Lemma by combining Anosov’s Stability Lemma with the Stable Manifold Theorem. In essence we prove a perturbed Stable Manifold Theorem. In order to deal with weakly hyperbolic orbits we use Pugh and Shub’s graph transform version of Pesin’s Stable Manifold Theorem. Normally, the contraction required to prove either Anosov’s Stability Lemma or the Stable Manifold Theorem, is derived from the hyperbolicity of a “supporting” invariant set. In fact neither of these proofs require this invariance; hyperbolic, or even pseudo-hyperbolic, families of pseudo-orbits are all that they require. This allows us to conclude the existence of shadowing orbits in the neighbourhood of “hyperbolic invariant sets” of numerical simulations of lowdimensional dynamical systems. In particular corresponding to any such numerical “hyperbolic invariant set”, there is a uniformly hyperbolic invariant set of the dynamical system itself.