ERGODIC THEORY AND DYNAMICS OF G-SPACES (with special emphasis on rigidity phenomena) Renato Feres Anatole Katok Contents Chapter 1. Introduction 5 1.1. Dynamics of group actions in mathematics and applications 5 1.2. Properties of groups relevant to dynamics 6 1.3. Rigidity phenomena 7 1.4. Rigid geometric structures 9 1.5. Preliminaries on Lie groups and lattices 10 Chapter 2. Basic ergodic theory 15 2.1. Measurable G-actions 15 2.2. Ergodicity and recurrence 16 2.3. Cocycles and related constructions 23 2.4. Reductions of principal bundle extensions 27 2.5. Amenable groups and amenable actions 30 Chapter 3. Groups actions and unitary representations 35 3.1. Spectral theory 35 3.2. Amenability and property T 41 3.3. Howe-Moore ergodicity theorem 44 Chapter 4. Main classes of examples 49 4.1. Homogeneous G-spaces 49 4.2. Automorphisms of compact groups and related examples 52 4.3. Isometric actions 54 4.4. Gaussian dynamical systems 56 4.5. Examples of actions obtained by suspension 57 4.6. Blowing up 58 Chapter 5. Smooth actions and geometric structures 59 5.1. Local properties 59 5.2. Actions preserving a geometric structure 60 5.3. Smooth actions of semisimple Lie groups 65 5.4. Dynamics, rigid structures, and the topology of M 68 Chapter 6. Actions of semisimple Lie groups and lattices of higher real-rank 71 6.1. Preliminaries 71 6.2. The measurable theory 71 6.3. Topological superrigidity 80 6.4. Actions on low-dimensional manifolds 82 6.5. Local differentiable rigidity of volume preserving actions 85 6.6. Global differentiable rigidity with standard models 88 6.7. Actions without invariant measures 89 3 4 CONTENTS Bibliography 93 CHAPTER 1 Introduction 1. Dynamics of group actions in mathematics and applications The theory of dynamical systems deals with properties of groups or semi-groups of transformations that are asymptotic in character, that is, that become apparent as “one goes to infinity in the group.” Since its beginnings in classical and statis- tical mechanics, the theory has been dedicated primarily to actions of the additive groups R and Z. These groups are typically interpreted as parametrizing time - the corresponding actions usually represent the time evolution of a system, respectively as a flow or by the iteration of an invertible transformation. Somewhat more recently, dynamical methods and ideas began to be applied to actions of more general groups. To cite a few examples, the work of G. Mackey shows an important role played by ergodic theory in the study of unitary represen- tations of general (second countable locally compact) groups. (See [Ma2]; see also [Va1] for the relevance of Mackey’s work to the foundations of quantum theory.) In probability theory, actions of Zn arise in the context of the statistical physics of ferromagnetic materials and the Lenz-Ising spin lattices. (See [Rue] and references therein.) A certain action of SL(2, R) on Teichmuller space plays a central role in the study of rational billiards (see survey [S-MT]), whereas the dynamics of homogeneous actions of unipotent and other subgroups of certain semisimple Lie groups has been applied with great success to problems in number theory. (See survey [S-KSS].) Also, the study of the geometry and topology of manifolds of nonpositive curvature, as well as the study of linear representations of lattices in semisimple Lie groups propelled the development of the ergodic theory of noncom- pact semisimple Lie groups and their lattice subgroups, a topic that will be explored in some detail in the present survey. Seminal work in this direction was done by Mostow, Margulis, and Furstenberg. See [Mos, Mar2, Z1]. According to the general scheme presented in [S-HK] we will consider actions of a group G on a set X that leave invariant some structure on X such as, say, a topology, a finite or infinite measure or a measure class, a smooth manifold structure, or a symplectic form. The various branches of dynamics correspond to putting the focus on each of these different structures, respectively: ergodic theory, topological, smooth, symplectic, holomorphic, algebraic (homogeneous and affine) dynamics and so on. Of course, these are not fully independent: for example, differentiable structure produces topology and symplectic systems preserve a smooth volume form and hence a canonical (Liouville) measure; similarly, homogeneous systems may preserve a Haar measure. 5 6 1. INTRODUCTION 2. Properties of groups relevant to dynamics a. Amenable vs. non-amenable groups. In the context of the general theory of dynamical systems, the acting groups should not be too big: locally compact second countable is a standard assumption. Following [S-HK] we will call actions of Z, Z+, R or R+ cyclic dynamical systems. Most of the surveys in this volume deal either exclusively or primarily with such systems. Certain general facts about cyclic dynamical systems that are discussed in this volume, particularly in [S-HK], either extend directly to, or have counterparts for, actions of locally compact second countable groups. Some topics that do not have a ready extension to this general class of groups as, for example, entropy theory and ergodic theorems, can be extended to certain classes of groups that share essential properties with the above “small” ones, such as amenability. (See Section 2.5 and [S-HK].) Surveys [S-B, S-LS, S-KSS] pay considerable attention to noncyclic dynam- ical systems although in the first two cases actions of amenable groups and semi- groups are primarily considered. In what follows we will restrict the discussion to the case of group, as opposed to semigroup, actions. There are various technical complications that arise in passing from invertible to non-invertible actions and in general the non-amenable non- invertible case has not been given sufficient attention as yet to warrant inclusion into a general survey like this one. The primary focus of the survey is on those aspects of the ergodic theory and differentiable dynamics of group actions that are most distinct from the theory for R and Z. Since ergodic theory for actions of general amenable groups share with R and Z many key properties, this survey will be concerned in large part with actions of non-amenable groups. Orbit equivalence provides a particularly compelling example: on the one hand, all finite measure–preserving ergodic actions of discrete amenable groups are orbit equivalent [CFW]; on the other hand, for certain groups, which are both “sufficiently large” and “rigid”, orbit equivalence essentially implies isomorphism (cf. Sections 6.2c and 6.2d). b. Complexity of the group structure. An underlying theme that runs through this survey is the influence that the greater complexity of the acting group itself has on the dynamical properties of the actions. That complexity presents itself in several ways. Amenability, for example, is associated with low complexity in the sense of moderate (volume) growth in the group, or in the limited “ways” or “directions” along which one can go out to infinity within the group. This can be expressed more precisely by the various notions of group boundary. (See [Kai] for a survey on the various probabilistic and algebraic notions of group boundaries.) The very existence of invariant probability measures for an action is an issue connected with properties of these boundaries, as will be seen later. (See Chapter 4, for a brief discussion of H. Furstenberg’s work on boundary actions, and Chapter 6, on the work by A. Nevo and R. Zimmer on the structure of actions of higher rank semisimple Lie groups without invariant measures.) Yet another indication of complexity in the group structure is seen in the way subgroups are “interlocked” within the group. For example, for a simple Lie group, dynamical properties of the restriction of the action to a maximal abelian R- diagonalizable subgroup (an R-split Cartan subgroup) goes a long way to determine properties of the action of the whole group. (The Howe-Moore ergodicity theorem 1.3. RIGIDITY PHENOMENA 7 illustrates this point well. See Section 3.3 of Chapter 2.) This is specially true when the R-split Cartan subgroup has rank (as an abelian group) two or greater, as will be seen in many of the rigidity phenomena for “higher-rank” actions discussed later in this survey. The theory of unitary group representations is an effective tool for relating the structure of a group with the dynamics and ergodic theory of its actions. Kazh- dan’s property T (Section 3.2) is a prominent example of a property defined by means of the unitary representations of the group that has considerable (although not fully understood as yet) ergodic theoretic implications. Property T is a very useful ingredient in the analysis of actions of more special classes of groups, such as semisimple Lie groups of real-rank greater than one. Amenability can also be formulated in terms of the unitary representations of the group, and it is, in fact, a natural opposite to property T . For example, the only amenable groups with property T are compact groups, which are trivial from the point of view of asymptotic behavior. Connected semisimple Lie groups of real-rank greater than one possess all the attributes of complexity described above: a complicated web of Cartan subgroups, responsible for a rich algebraic structure of the boundary at infinity (Bruhat-Tits buildings), Kazhdan’s property T , and in general a theory of irreducible unitary representations that has very different properties compared to amenable groups. Among the non-amenable groups, higher real-rank semisimple Lie groups and their lattices are the ones for which both measurable and smooth dynamics are best understood.