Non-stationary dynamical systems; Shadowing theorem and some applications

Abstract

In the present paper, we mean a sequence of maps along a sequence of spaces by a non-stationary dynamical system. We use an Anosov family as a generalization of an Anosov map, which is a sequence of diffeomorphisms along a sequence of compact Riemannian manifolds, so that the tangent bundles split into expanding and contracting subspaces, with uniform bounds for the contraction and the expansion. Also, we introduce the shadowing property on non-stationary dynamical systems. Then, we prepare the necessary conditions for the existence of the shadowing property to prove the shadowing theorem in nonstationary dynamical systems. The shadowing theorem is a known result in dynamical systems, which states that any dynamical system with a hyperbolic structure has the shadowing property. Here, we prove that the shadowing theorem is established on any invariant Anosov family in a non-stationary dynamical system. Then, as in some applications of the shadowing theorem, we check the stability of Anosov families, and also we peruse the stability of isolated invariant Anosov families in non-stationary dynamical systems.