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Anosov diffeomorphisms. (English) Zbl 0203.26101
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 Other theorems concerning topological classification have in the meantime been superseded by the result of Franks and Newhouse that every Anosov diffeomorphism of codimension one (i.e. such that $$\dim E^u=1)$$ is topologically conjugate to some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.)