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On Anosov diffeomorphisms with asymptotically conformal periodic data

Published online by Cambridge University Press:  01 February 2009

BORIS KALININ  and
VICTORIA SADOVSKAYA
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)
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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
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 117 - 136
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Burns, K. and Wilkinson, A.. A note on stable holonomy between centers. Preprint.Google Scholar
[2]Fang, Y.. Smooth rigidity of uniformly quasiconformal Anosov flows. Ergod. Th. & Dynam. Sys. 24(6) (2004), 1937–1959.CrossRefGoogle Scholar
[3]Gogolev, A. and Guysinsky, M.. C 1-differentiable conjugacy for Anosov diffeomorphisms on three dimensional torus. Discrete Contin. Dyn. Syst. 22(1–2) (2008), 183–200.Google Scholar
[4]Hurder, S. and Katok, A.. Ergodic theory and Weil measures for foliations. Ann. of Math. (2) 126(2) (1987), 221–275.CrossRefGoogle Scholar
[5]Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds. Springer, New York, 1977.CrossRefGoogle Scholar
[6]Journé, J.-L.. A regularity lemma for functions of several variables. Rev. Mat. Iberoamericana 4(2) (1988), 187–193.CrossRefGoogle Scholar
[7]Kalinin, B. and Sadovskaya, V.. On local and global rigidity of quasiconformal Anosov diffeomorphisms. J. Inst. Math. Jussieu 2(4) (2003), 567–582.CrossRefGoogle Scholar
[8]Kalinin, B. and Sadovskaya, V.. Global rigidity for totally nonsymplectic Anosov ℤk actions. Geom. Topol. 10 (2006), 929–954.CrossRefGoogle Scholar
[9]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, New York, 1995.CrossRefGoogle Scholar
[10]de la Llave, R.. Invariants for smooth conjugacy of hyperbolic dynamical systems II. Comm. Math. Phys. 109 (1987), 368–378.CrossRefGoogle Scholar
[11]de la Llave, R.. Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems. Comm. Math. Phys. 150 (1992), 289–320.CrossRefGoogle Scholar
[12]de la Llave, R.. Rigidity of higher-dimensional conformal Anosov systems. Ergod. Th. & Dynam. Sys. 22(6) (2002), 1845–1870.CrossRefGoogle Scholar
[13]de la Llave, R. and Windsor, A.. Livsic theorem for non-commutative groups including groups of diffeomorphisms, and invariant geometric structures. Preprint.Google Scholar
[14]de la Llave, R.. Further rigidity properties of conformal Anosov systems. Ergod. Th. & Dynam. Sys. 24(5) (2004), 1425–1441.CrossRefGoogle Scholar
[15]de la Llave, R. and Moriyón, R.. Invariants for smooth conjugacy of hyperbolic dynamical systems IV. Comm. Math. Phys. 116 (1988), 185–192.CrossRefGoogle Scholar
[16]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Ann. Math. 122 (1985), 509–539.CrossRefGoogle Scholar
[17]Pollicott, M.. C k-rigidity for hyperbolic flows II. Israel J. Math. 69 (1990), 351–360.CrossRefGoogle Scholar
[18]Rodriguez Hertz, F.. Global rigidity of certain abelian actions by toral automorphisms. J. Modern Dyn. 1(3) (2007), 425–442.CrossRefGoogle Scholar
[19]Sadovskaya, V.. On uniformly quasiconformal Anosov systems. Math. Res. Lett. 12(3) (2005), 425–441.CrossRefGoogle Scholar
[20]Sun, W. and Wang, Z.. Lyapunov exponents of hyperbolic measures and hyperbolic periodic points. Preprint.Google Scholar
[21]Kalinin, B.. Livsic theorem for matrix cocycles. Preprint.Google Scholar