Anosov diffeomorphisms. (English) Zbl 0203.26101

Avez, A.

Topol. Dynamics, Int. Symp. Colorado State Univ. 1967, 17-51 (1968).

A diffeomorphism \(f\) of a compact manifold \(M\) is an Anosov diffeomorphism if \(TM\) splits into a Whitney sum \(E^u\otimes E^d\) invariant under \(Df\) such that \(Df\mid E^d\) and \(Df^{-1}\mid E^u\) are contracting. This report first summarizes some basic theorems and examples, and then gives proofs of the author’s results for the case that \(M\) possesses a continuous invariant measure \(\mu > 0\). In particular, in this case the universal covering of \(M\) is a Euclidean space. Other theorems concerning topological classification have in the meantime been superseded by the result of Franks and Newhouse that every Anosov diffeomorphism of codimension one (i.e. such that \(\dim E^u=1)\) is topologically conjugate to some toral automorphism.
[For the entire collection see Zbl 0192.31103.]

Reviewer: K. Sigmund

MSC:

37D20

Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Citations:

Zbl 0192.31103