Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori
Andrey Gogolev
Journal of Modern Dynamics 2008, 2(4): 645-700 doi: 10.3934/jmd.2008.2.645
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: flag of foliations hyperbolic automorphism absolutely continuous measure regularity of holonomy map smooth conjugacy Anosov diffeomorphism moduli of smooth conjugacy periodic data minimal foliation de la Llave counterexample
$C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus
Andrey Gogolev Misha Guysinsky
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 183-200 doi: 10.3934/dcds.2008.22.183
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.
keywords: Anosov diffeomorphism smooth conjugacy periodic data.
Partially hyperbolic diffeomorphisms with compact center foliations
Andrey Gogolev
Journal of Modern Dynamics 2011, 5(4): 747-769 doi: 10.3934/jmd.2011.5.747
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:
  1. 1. the center foliation of $f$ has codimension 2, or
  2. 2. the center leaves of $f$ are simply connected leaves and the unstable foliation of $f$ is one-dimensional.
keywords: Reeb stability skew product compact foliation Anosov homeomorphism Wada Lakes Partially hyperbolic diffeomorphism accessible boundary holonomy.

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