An Ergodic-Theoretical Approach to the Chaotic Behaviour of Dynamical Systems

Abstract

In the present paper we shall survey a series of results concerning the chaotic behaviours of one dimensional maps from an ergodic theoretical aspect, which has a far origin in a meeting on the topological entropy held at RIMS a little more than ten years ago. In recent years the chaotic behaviour of dynamical systems is a common concern not only among various branches of mathematics but also among various branches of sciences and engineerings. For instance, statistical physicists seem to have accepted it as one of the general behaviours in non equilibrium statistical mechanics. Especially, Feigenbaum's and other critical phenomena in bifurcation diagrams of dynamical systems are intensively studied among them. As to the statistical mechanics the close relationship was established in 1970's between ergodic theory and equilibrium statistical mechanics since Sinai [22]. One may say that the theory of Axiom A diffeomorphisms and expanding maps is isomorphic to the theory of the equilibrium statistical mechanics of one dimensional lattice systems and the introduction of the notion of Gibbs measures has brought several deep results such as [3]. The current attention to the chaotic behaviour may be said to have started with the papers of Ruelle-Takens [21], May [14], Li-Yorke [11], Feigenbaum [34] et al. The Veda attractor called Japanese attractor by D. Ruelle for the periodically forced Duffing equation was already observed by Y. Ueda in 1960's but it did not draw any attention till recent years ([33]). The existence of the complicated behaviours of simple dynamical systems in itself was known since