Dynamical properties of selfmaps on infra-nilmanifolds.

Expanding maps and Anosov diffeomorphisms are important types of dynamical systems since they were among the first examples with structural stability and chaotic behavior. Every closed manifold admitting an expanding map is homeomorphic to an infra-nilmanifold and it is conjectured that the same is true for manifolds admitting an Anosov diffeomorphism. This motivates the research of expanding maps and Anosov diffeomorphisms on infra-nilmanifolds.

Although, up to homeomorphism, infra-nilmanifolds are the only closed manifolds supporting an expanding map, not every infra-nilmanifold admits an expanding map. Similarly the existence of an Anosov diffeomorphism on an infra-nilmanifold puts strong conditions on its fundamental group. This dissertation studies which infra-nilmanifolds admit an expanding map or an Anosov diffeomorphism. Because of the algebraic nature of infra-nilmanifolds, these questions are translated into studying the group morphisms of their fundamental groups, which are exactly the almost-Bieberbach groups. The main results of this essay give algebraic methods for deciding whether a given infra-nilmanifold admits an expanding map or an Anosov diffeomorphism.

The proofs in this dissertation combine methods from different branches in mathematics, including (geometric) group theory, number theory, Lie algebras, linear algebraic groups and representation theory of finite groups. The first part of this thesis gives the necessary background about the definitions and results in these areas which are needed throughout the following chapters. The emphasis of this first part is on self-maps of infra-nilmanifolds and the relation to expanding maps and Anosov diffeomorphisms.

The second part focuses on the situation of expanding maps. The main result gives an algebraic criterion to decide whether an infra-nilmanifold admits an expanding map or not. This criterion only depends on the covering Lie group and more specific on the existence of a positive grading on the corresponding Lie algebra. The proof of this result consists of two steps which use different methods. The first step deals with group morphisms of commensurable nilpotent groups. These techniques are also useful for determining the periodic points for a big class of self-maps on infra-nilmanifolds. The second step involves gradings on Lie algebras and the relation to automorphisms on these Lie algebras.

The third part of the thesis discusses the existence of Anosov diffeomorphisms on infra-nilmanifolds. Since every Anosov diffeomorphism on an infra-nilmanifold can be lifted to a covering nilmanifold, a first step is to understand which nilmanifolds admit an Anosov diffeomorphism or equivalently to study Anosov Lie algebras. A new method for constructing Anosov Lie algebras is given which answers many open existence questions about these maps. The new examples include Anosov diffeomorphisms of minimal signature, Anosov Lie algebras of minimal type and a nilmanifold admitting an Anosov diffeomorphism but no expanding map. Once we understand which nilmanifolds in a certain class of infra-nilmanifolds admit an Anosov diffeomorphism, the next step is to study the infra-nilmanifolds in this same class. This essay gives an algebraic description of the infra-nilmanifolds modeled on free nilpotent Lie groups which admit an Anosov diffeomorphism.

In the final chapter, we give some directions for future research by stating open questions which originate from this dissertation. This chapter also contains new results connecting this dissertation to other recent research and describes some methods for tackling these problems.