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On Anosov diffeomorphisms with asymptotically conformal periodic data

Published online by Cambridge University Press:  01 February 2009

BORIS KALININ
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA (email: kalinin@jaguar1.usouthal.edu, sadovska@jaguar1.usouthal.edu)
VICTORIA SADOVSKAYA
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA (email: kalinin@jaguar1.usouthal.edu, sadovska@jaguar1.usouthal.edu)

Abstract

We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C1+β smoothness of the Anosov splitting, and C1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume-preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of 𝕋4; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

Type
Research Article
Copyright
Copyright © 2008 Cambridge University Press

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