## Journals

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### Open Access Journals

JMD

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.

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.

DCDS

We consider two $C^2$ Anosov diffeomorphisms in a $C^1$ neighborhood of a linear hyperbolic automorphism of
three dimensional torus with real spectrum. We prove that they are $C^{1+\nu}$ conjugate if and only if the
differentials of the return maps at corresponding periodic points have the same eigenvalues.

JMD

Let $f\colon M\to M$ be a partially hyperbolic diffeomorphism such
that all of its center leaves are compact. We prove that Sullivan's
example of a circle foliation that has arbitrary long leaves cannot be
the center foliation of $f$. This is proved by thorough study of the
accessible boundaries of the center-stable and the center-unstable
leaves.

Also we show that a finite cover of $f$ fibers over an Anosov toral automorphism if one of the following conditions is met:

Also we show that a finite cover of $f$ fibers over an Anosov toral automorphism if one of the following conditions is met:

- 1. the center foliation of $f$ has codimension 2, or
- 2. the center leaves of $f$ are simply connected leaves and the unstable foliation of $f$ is one-dimensional.

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