Abstract. This is a set of notes of a series of lectures given by the author on the International conference on Dynamical Systems and Ergodic Theory, Katseveli 2000 where we will demonstrate a various techniques and methods used in the study of interval maps. In particular, we will focus on the negative Schwarzian derivative condition and on the density of Axiom A maps.

INTRODUCTION

The aim of this paper is to introduce the Structural stability problem and to demonstrate different techniques and methods of the theory of Dynamical systems on examples arising from this problem. Though these methods usually are quite technical, we will try to avoid giving all the details and instead we will try to demonstrate the underlying ideas.

The Structural stability problem deals with the following question: What dynamical systems do not change their topological behaviour under small perturbations? Such dynamical systems are called structurally stable:

Definition 1. Let M be a manifold. The Ckmap f : M → M is called Ck structurally stable if there is ∊ > 0 such that any Ckmap g : M → M, ||f — g||Ck < ∊, is topologically conjugate to f (i.e. there exists a homeomorphism h of M such that g º h = h º f).