Localization of the Chain Recurrent set using Shape theory and Symbolical Dynamics

Abstract

The main aim of this paper is localization of the chain recurrent set in shape theoretical framework. Namely, using the intrinsic approach to shape from [1] we present a result which claims that under certain conditions the chain recurrent set preserves local shape properties. We proved this result in [2] using the notion of a proper covering. Here we give a new proof using the Lebesque number for a covering and verify this result by investigating the symbolical image of a couple of systems of differential equations following [3].

A dynamical approach to shape

“…The proof of the above lemma is an appropriate adjustment of the proof of a lemma given in the paper [22]. In the following section we will give some insights to the question from [23]: "Is every shape equivalence f : M Ñ X a strong shape equivalence?…”