Journal of Modern Dynamics (JMD)

Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori

Pages: 645 - 700, Issue 4, October 2008      doi:10.3934/jmd.2008.2.645

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Andrey Gogolev - Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, United States (email)

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.

Keywords:  Anosov diffeomorphism, hyperbolic automorphism, smooth conjugacy, moduli of smooth conjugacy, periodic data, de la Llave counterexample, minimal foliation, flag of foliations, absolutely continuous measure, regularity of holonomy map
Mathematics Subject Classification:  Primary: 37C15, 37D20; Secondary: 37D30, 34D30, 34D10, 37C25, 37C40

Received: April 2008;      Revised: May 2008;      Available Online: October 2008.