Rigidity for Some Cases of Anosov Endomorphisms of Torus

TL;DR

This paper establishes smooth conjugacy results for certain Anosov endomorphisms on tori, including conditions under which these maps are smoothly conjugate to their linearizations, advancing understanding of rigidity in dynamical systems.

Contribution

It proves smooth conjugacy between specific classes of Anosov endomorphisms and their linearizations, extending rigidity results under various regularity and periodic data conditions.

Findings

Smooth conjugacy between non-necessarily special Anosov endomorphisms and their linearizations.

Rigidity results for strongly special $C^{ abla}-$Anosov endomorphisms of $ op^2$.

Local rigidity of linear Anosov endomorphisms of $d$-torus for $d \\geq 3$.

Abstract

We obtain smooth conjugacy between non-necessarily special Anosov endomorphisms in the conservative case. Among other results, we prove that a strongly special $C^{\infty}-$Anosov endomorphism of $\mathbb{T}^2$ and its linearization are smoothly conjugated since they have the same periodic data. Assuming that for a strongly special $C^{\infty}-$Anosov endomorphism of $\mathbb{T}^2$ every point is regular (in Oseledec's Theorem sense), then we obtain again smooth conjugacy with its linearization. We also obtain some results on local rigidity of linear Anosov endomorphisms of $d-$torus, where $d \geq 3,$ under periodic data assumption. The study of differential equations defined on invariant leaves plays an important role in rigidity problems such as those treated here.