{"title":"A characterization of Anosov rational forms in nilpotent Lie algebras associated to graphs","authors":"Jonas Deré, Thomas Witdouck","doi":"10.1007/s00605-024-01978-8","DOIUrl":null,"url":null,"abstract":"<p>Anosov diffeomorphisms are an important class of dynamical systems with many peculiar properties. Ever since they were introduced in the sixties, it has been an open question which manifolds can admit such diffeomorphisms, where tori of dimension greater than or equal to two are the typical examples. It is conjectured that the only manifolds supporting an Anosov diffeomorphism are finitely covered by a nilmanifold, a type of manifold closely related to rational nilpotent Lie algebras. In this paper, we study the existence of Anosov diffeomorphisms for a large class of these nilpotent Lie algebras, namely the ones that can be realized as a rational form in a Lie algebra associated to a graph. From a given simple undirected graph, one can construct a complex <i>c</i>-step nilpotent Lie algebra, which in general contains different non-isomorphic rational forms, as described by the authors in previous work. We determine precisely which forms correspond to a nilmanifold admitting an Anosov diffeomorphism, leading to the first class of complex nilpotent Lie algebras having several non-isomorphic rational forms and for which all the ones that are Anosov are described. In doing so, we put a new perspective on certain classifications in low dimensions and correct a false result in the literature.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01978-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Anosov diffeomorphisms are an important class of dynamical systems with many peculiar properties. Ever since they were introduced in the sixties, it has been an open question which manifolds can admit such diffeomorphisms, where tori of dimension greater than or equal to two are the typical examples. It is conjectured that the only manifolds supporting an Anosov diffeomorphism are finitely covered by a nilmanifold, a type of manifold closely related to rational nilpotent Lie algebras. In this paper, we study the existence of Anosov diffeomorphisms for a large class of these nilpotent Lie algebras, namely the ones that can be realized as a rational form in a Lie algebra associated to a graph. From a given simple undirected graph, one can construct a complex c-step nilpotent Lie algebra, which in general contains different non-isomorphic rational forms, as described by the authors in previous work. We determine precisely which forms correspond to a nilmanifold admitting an Anosov diffeomorphism, leading to the first class of complex nilpotent Lie algebras having several non-isomorphic rational forms and for which all the ones that are Anosov are described. In doing so, we put a new perspective on certain classifications in low dimensions and correct a false result in the literature.
阿诺索夫衍射是一类重要的动力系统,具有许多奇特的性质。自从阿诺索夫衍射在六十年代被提出以来,哪些流形可以容纳这种衍射一直是一个悬而未决的问题,其中维度大于或等于二的环是典型的例子。有人猜想,唯一支持阿诺索夫差分的流形是由无流形有限覆盖的,无流形是一种与有理无势列阵密切相关的流形。在本文中,我们研究了一大类此类无穷烈度代数的阿诺索夫差分形的存在性,即那些可以在与图相关联的烈度代数中以有理形式实现的无穷烈度代数。从给定的简单无向图中,我们可以构造出一个复杂的 c 阶零势列代数,正如作者在之前的工作中所描述的,它一般包含不同的非同构有理形式。我们精确地确定了哪些形式对应于容许阿诺索夫差分变形的无芒点,从而产生了第一类具有多个非同构有理形式的复零势列代数,并描述了其中所有阿诺索夫形式。这样,我们就为低维度的某些分类提供了一个新的视角,并纠正了文献中的一个错误结果。