Topological ergodic theory and mean rotation

Abstract

Listen

Translate article

Let H 0 ( M n , μ ) {\mathcal {H}_0}({M^n},\mu ) denote the set of all homeomorphisms of a compact manifold M n {M^n} that preserve a locally positive nonatomic Borel probability measure μ \mu and are isotopic to the identity. The notion of the mean rotation vector for a torus homeomorphism has been extended by Fathi to a continuous map θ \theta on H 0 ( M n , μ ) {\mathcal {H}_0}({M^n},\mu ) . We show that any abstract ergodic behavior typical for automorphisms of ( M n , μ ) ({M^n},\mu ) as a Lebesgue space is also typical not only in H 0 ( M n , μ {\mathcal {H}_0}({M^n},\mu but also in each closed subset of constant θ \theta . By typical we mean dense G δ {G_\delta } in the appropriate space. Weak mixing is an example of such a typical abstract ergodic behavior. This contrasts sharply with a deep result of the KAM theory that for some rotation vectors v → \overrightarrow v , there is an open neighborhood of rotation by v → \overrightarrow v , in the space of smooth volume preserving n n -torus diffeomorphisms with θ = v → \theta = \overrightarrow v , where each diffeomorphism in the open set is conjugate to rotation by v → \overrightarrow v (and hence cannot be weak mixing).