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Abstract: Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study thesmooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$.The main result establishes $C^{1+ u}$ regularity of the conjugacy betweentwo Anosov systems with the same periodic eigenvalue data. We assume that thesesystems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$with simple real spectrum and that they satisfy a natural transitivityassumption on certain intermediate foliations.We elaborate on the example of de la Llave of two Anosov systems on $\mathbbT^4$ with the same constant periodic eigenvalue data that are only H\-olderconjugate. We show that these examples exhaust all possible ways to perturb$C^{1+ u}$ conjugacy class without changing periodic eigenvalue data. Also wegeneralize these examples to majority of reducible toral automorphisms as wellas to certain product diffeomorphisms of $\mathbb T^4$ $C^1$-close to theoriginal example.



Author: Andrey Gogolev

Source: https://arxiv.org/







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