Stability Versus Hyperbolicity in Dynamical and Iterated Function Systems

Abstract

In this paper we investigate a certain notion of stability, for one function or for iterated function systems, and discuss why this notion can be a good extension and complement to the notion of hyperbolicity. This last notion is very well-known in the literature and plays an important role in the investigation of the dynamical behavior of a system. The main result is that although some classical sets of functions like the stable Lipschitz functions are conjugate to hyperbolic functions there exist continuous stable functions which are not conjugate to hyperbolic functions. A sufficient condition for not being conjugate to a hyperbolic function is given.