Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori

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

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 HÃ¶lder 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.