### 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 non-stationary dynamical system and we use 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 neccessary conditions for the existence of the shadowing property to prove the Shadowing Theorem in non-stationary dynamical systems. Shadowing Theorem is a known result in dynamical systems which states any dynamical system with 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 some applications of Shadowing Theorem, we check stability of Anosov families and also we peruse stability of isolated invariant Anosov families in non-stationary dynamical systems.

of spaces by non-stationary dynamical system and we use 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 neccessary conditions for the existence of the shadowing property to prove the Shadowing Theorem in non-stationary dynamical systems. Shadowing Theorem is a known result in dynamical systems which states any dynamical system with 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 some applications of Shadowing Theorem, we check stability of Anosov families and also we peruse stability of isolated invariant Anosov families in non-stationary dynamical systems.

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