Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori

Andrey Gogolev

doi:10.3934/jmd.2008.2.645

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2008, Volume 2, Issue 4: 645-700. Doi: 10.3934/jmd.2008.2.645

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Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori

Andrey Gogolev^1,

1.
Department of Mathematics, Pennsylvania State University, University Park, PA, 16802

Received: March 31, 2008

Revised: April 30, 2008

Published: September 2008

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Abstract

Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$.

The main result establishes $C^{1+\nu}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations.

We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same constant periodic eigenvalue data that are only Hölder conjugate. We show that these examples exhaust all possible ways to perturb a $C^{1+\nu}$ conjugacy class without changing any periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of $\mathbb T^4$ $C^1$-close to the original example.

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Mathematics Subject Classification: Primary: 37C15, 37D20; Secondary: 37D30, 34D30, 34D10, 37C25, 37C40.

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