Michael Shub (IBM) Mathematics Department Colloquium

Order out of Chaos? Some recent examples.

Abstract. One of the hallmarks of chaotic dynamical systems is that the long term future is difficult to predict deterministically because it depends very sensitively on initial conditions. In these circumstances one may still be able to make statistical predictions. Some of the earliest results in this direction are Hopf's theorem on the ergodicity of the geodesic flow on compact surfaces of constant negative curvature in the 1930's and Anosov's generalization on the ergodicity of uniformly hyperbolic systems in the 1960's. Here we discuss some recent generalizations of the theorems of Hopf and Anosov and the resolution of some long standing problems by Dolgopyat-Pesin and Rodriguez-Hertz. Some comments will be made contrasting the topological classification and ergodic theory of these systems. I intend the talk to stay on an elementary level with the main examples given by linear automorphisms of tori. Let Abe an element of $SL(n,\mathbb Z)$ the n by n matrices with integer entries and determinant one. Then A defines a linear map of ${\mathbb R}^n$. Since the integer lattice ${\mathbb Z}^n$ is preserved by A, A induces a diffeomorphism of the n-torus, ${\mathbb T}^n$, where ${\mathbb T}^n$ is considered as ${\mathbb R}^n$ mod ${\mathbb Z}^n$.