Stability theorems and hyperbolicity in dynamical systems

Abstract

We present an approach to proving structural stability and semi-stability theorems for diffeomorphisms and flows using the idea of shadowing an ∈-chain. We treat the cases when the whole manifold is hyperbolic and when the chain recurrent set is hyperbolic and the strong transversality condition is satisfied. The last two sections discuss the progress made on the converse to the structural stability theorem and a criterion for hyperbolicity. © 1977 Rocky Mountain Mathematics Consortium.