Mathematical Research Letters

Volume 18 (2011)

Number 5

Local Rigidity For Anosov Automorphisms

Pages: 843 – 858

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n5.a4

Authors

Andrey Gogolev (Department of Mathematics, University of Texas at Austin)

Boris Kalinin (Department of Mathematics & Statistics, University of South Alabama, Mobile, Ala. U.S.A.)

Rafael de la Llave (Department of Mathematics, University of Texas at Austin, Austin)

Victoria Sadovskaya (Department of Mathematics & Statistics, University of South Alabama, Mobile, Ala., U.SA.)

Abstract

We consider an irreducible Anosov automorphism $L$ ofa torus $\T^d$ such thatno three eigenvalues have the same modulus. We show that $L$is locally rigid, that is, $L$ is $C^{1+\text{Hölder}}$ conjugateto any $C^1$-small perturbation $f$ such that the derivative$D_pf^n$ is conjugate to $L^n$ whenever $f^np=p$.We also prove that toral automorphisms satisfying theseassumptions are generic in $SL(d,\Z)$.Examples constructed in the Appendix show the importanceof the assumption on the eigenvalues.

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