Preprint des Projektes: SFB 343: Diskrete Strukturen in der Mathematik

Theorie und Numerik von Aufgaben der linearen Algebra und diskreter dynamischer Systeme

00-072 Wolf-Jürgen Beyn.
A Note on Symbolic Dynamics near Connecting Orbits of Maps

In this note we consider the dynamics of a diffeomorphism on a maximal, invariant set in the neighborhood of a transversal homoclinic orbit. It is shown that the map conjugates to a topological Markov chain on a finite number of symbols defined by a special transition matrix. The theorem collects in a unified way various well-known results of Smale, Shilnikov,Palmer and others on symbolic dynamics. Generalizations of the result are indicated for finite sets of hyperbolic fixed points with a given finite collection of transversal connecting orbits between them.