DOCSLIB.ORG

ERGODIC THEORY and DYNAMICS of G-SPACES (With Special Emphasis on Rigidity Phenomena)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
ERGODIC THEORY and DYNAMICS of G-SPACES (With Special Emphasis on Rigidity Phenomena) 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.
Load more

Recommended publications
  • Contents 1. Introduction 1 2. Cones in Vector Spaces 2 2.1. Ordered Vector Spaces 2 2.2
    ORDERED VECTOR SPACES AND ELEMENTS OF CHOQUET THEORY (A COMPENDIUM) S. COBZAS¸ Contents 1. Introduction 1 2. Cones in vector spaces 2 2.1. Ordered vector spaces 2 2.2. Ordered topological vector spaces (TVS) 7 2.3. Normal cones in TVS and in LCS 7 2.4. Normal cones in normed spaces 9 2.5. Dual pairs 9 2.6. Bases for cones 10 3. Linear operators on ordered vector spaces 11 3.1. Classes of linear operators 11 3.2. Extensions of positive operators 13 3.3. The case of linear functionals 14 3.4. Order units and the continuity of linear functionals 15 3.5. Locally order bounded TVS 15 4. Extremal structure of convex sets and elements of Choquet theory 16 4.1. Faces and extremal vectors 16 4.2. Extreme points, extreme rays and Krein-Milman's Theorem 16 4.3. Regular Borel measures and Riesz' Representation Theorem 17 4.4. Radon measures 19 4.5. Elements of Choquet theory 19 4.6. Maximal measures 21 4.7. Simplexes and uniqueness of representing measures 23 References 24 1. Introduction The aim of these notes is to present a compilation of some basic results on ordered vector spaces and positive operators and functionals acting on them. A short presentation of Choquet theory is also included. They grew up from a talk I delivered at the Seminar on Analysis and Optimization. The presentation follows mainly the books [3], [9], [19], [22], [25], and [11], [23] for the Choquet theory. Note that the first two chapters of [9] contains a thorough introduction (with full proofs) to some basics results on ordered vector spaces.
    [Show full text]
  • Cartan and Iwasawa Decompositions in Lie Theory
    CARTAN AND IWASAWA DECOMPOSITIONS IN LIE THEORY SUBHAJIT JANA Abstract. In this article we will be discussing about various decom- positions of semisimple Lie algebras, which are very important to un- derstand their structure theories. Throughout the article we will be assuming existence of a compact real form of complex semisimple Lie algebra. We will start with describing Cartan involution and Cartan de- compositions of semisimple Lie algebras. Iwasawa decompositions will also be constructed at Lie group and Lie algebra level. We will also be giving some brief description of Iwasawa of semisimple groups. 1. Introduction The idea for beginning an investigation of the structure of a general semisimple Lie group, not necessarily classical, is to look for same kind of structure in its Lie algebra. We start with a Lie algebra L of matrices and seek a decomposition into symmetric and skew-symmetric parts. To get this decomposition we often look for the occurrence of a compact Lie algebra as a real form of the complexification LC of L. If L is a real semisimple Lie algebra, then the use of a compact real form of LC leads to the construction of a 'Cartan Involution' θ of L. This involution has the property that if L = H ⊕ P is corresponding eigenspace decomposition or 'Cartan Decomposition' then, LC has a compact real form (like H ⊕iP ) which generalize decomposition of classical matrix algebra into Hermitian and skew-Hermitian parts. Similarly if G is a semisimple Lie group, then the 'Iwasawa decomposition' G = NAK exhibits closed subgroups A and N of G such that they are simply connected abelian and nilpotent respectively and A normalizes N and multiplication K ×A×N ! G is a diffeomorphism.
    [Show full text]
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
    [Show full text]
  • (Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
    A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding.
    [Show full text]
  • Arxiv:1404.0456V2 [Math.DS] 25 Aug 2015 9.3
    ON DENSITY OF ERGODIC MEASURES AND GENERIC POINTS KATRIN GELFERT AND DOMINIK KWIETNIAK Abstract. We provide conditions which guarantee that ergodic measures are dense in the simplex of invariant probability measures of a dynamical sys- tem given by a continuous map acting on a Polish space. Using them we study generic properties of invariant measures and prove that every invariant measure has a generic point. In the compact case, density of ergodic measures means that the simplex of invariant measures is either a singleton of a measure concentrated on a single periodic orbit or the Poulsen simplex. Our properties focus on the set of periodic points and we introduce two concepts: closeability with respect to a set of periodic points and linkability of a set of periodic points. Examples are provided to show that these are independent properties. They hold, for example, for systems having the periodic specification prop- erty. But they hold also for a much wider class of systems which contains, for example, irreducible Markov chains over a countable alphabet, all β-shifts, all S-gap shifts, C1-generic diffeomorphisms of a compact manifold M, and certain geodesic flows of a complete connected negatively curved manifold. Contents 1. Introduction 2 2. Preliminaries 5 3. Symbolic dynamics and examples 8 3.1. S-gap shifts 9 3.2. β-shifts 10 4. Closeability and approximability of ergodic measures 10 5. Linkability 13 6. Generic points 17 7. Proof of Theorem 1.1 20 8. Applications 22 8.1. C1-generic diffeomorphisms 22 8.2. Flows 23 9. Counterexamples 24 9.1.
    [Show full text]
  • Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
    Published in: Journal of Geometric Mechanics doi:10.3934/jgm.2017014 Volume 9, Number 3, September 2017 pp. 335{390 GEOMETRY OF MATRIX DECOMPOSITIONS SEEN THROUGH OPTIMAL TRANSPORT AND INFORMATION GEOMETRY Klas Modin∗ Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden Abstract. The space of probability densities is an infinite-dimensional Rie- mannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher{Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry|the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multi- variate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher{Rao geometries are discussed. The link to matrices is obtained by considering OMT and information ge- ometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent descrip- tion of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decomposi- tions (smooth and linear category), entropy gradient flows, and the Fisher{Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions.
    [Show full text]
  • Measure and Integration
    ¦ Measure and Integration Man is the measure of all things. — Pythagoras Lebesgue is the measure of almost all things. — Anonymous ¦.G Motivation We shall give a few reasons why it is worth bothering with measure the- ory and the Lebesgue integral. To this end, we stress the importance of measure theory in three different areas. ¦.G.G We want a powerful integral At the end of the previous chapter we encountered a neat application of Banach’s fixed point theorem to solve ordinary differential equations. An essential ingredient in the argument was the observation in Lemma 2.77 that the operation of differentiation could be replaced by integration. Note that differentiation is an operation that destroys regularity, while in- tegration yields further regularity. It is a consequence of the fundamental theorem of calculus that the indefinite integral of a continuous function is a continuously differentiable function. So far we used the elementary notion of the Riemann integral. Let us quickly recall the definition of the Riemann integral on a bounded interval. Definition 3.1. Let [a, b] with −¥ < a < b < ¥ be a compact in- terval. A partition of [a, b] is a finite sequence p := (t0,..., tN) such that a = t0 < t1 < ··· < tN = b. The mesh size of p is jpj := max1≤k≤Njtk − tk−1j. Given a partition p of [a, b], an associated vector of sample points (frequently also called tags) is a vector x = (x1,..., xN) such that xk 2 [tk−1, tk]. Given a function f : [a, b] ! R and a tagged 51 3 Measure and Integration partition (p, x) of [a, b], the Riemann sum S( f , p, x) is defined by N S( f , p, x) := ∑ f (xk)(tk − tk−1).
    [Show full text]
  • Conditions for Choquet Integral Representation of the Comonotonically Additive and Monotone Functional
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 282 (2003) 201–211 www.elsevier.com/locate/jmaa Conditions for Choquet integral representation of the comonotonically additive and monotone functional Yasuo Narukawa a,∗ and Toshiaki Murofushi b a Toho Gakuen, 3-1-10 Naka, Kunitachi, Tokyo 186-0004, Japan b Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259 Nagatuta, Midori-ku, Yokohama 226-8502, Japan Received 25 December 2000 Submitted by J. Horvath Abstract If the universal set X is not compact but locally compact, a comonotonically additive and mon- otone functional (for short c.m.) on the class of continuous functions with compact support is not represented by one Choquet integral, but represented by the difference of two Choquet integrals. The conditions for which a c.m. functional can be represented by one Choquet integral are discussed. 2003 Elsevier Science (USA). All rights reserved. Keywords: Nonadditive measure; Fuzzy measure; Cooperative game; Choquet integral; Comonotonic additivity 1. Introduction The Choquet integral with respect to a nonadditive measure is one of the nonlinear functionals defined on the class B of measurable functions on a measurable space (X, B). It was introduced by Choquet [1] in potential theory with the concept of capacity. Then, in the field of economic theory, it has been used for utility theory [17], and has been used for image processing and recognition [4,5], in the context of fuzzy measure theory [9,20]. Essential properties characterizing this functional are comonotonic additivity and monotonicity.
    [Show full text]
  • An Asymptotic Result on the A-Component in Iwasawa Decomposition
    Unspecified Journal Volume 00, Number 0, Pages 000–000 S ????-????(XX)0000-0 AN ASYMPTOTIC RESULT ON THE A-COMPONENT IN IWASAWA DECOMPOSITION HUAJUN HUANG AND TIN-YAU TAM MAY 12, 2006 Abstract. Let G be a real connected semisimple Lie group. For each v0, v, g ∈ G, we prove that lim [a(v0gmv)]1/m = s−1 · b(g), m→∞ where a(g) denotes the a-component in the Iwasawa decomposition of g = kan and b(g) ∈ A+ denotes the unique element that conjugate to the hyperbolic component in the complete multiplicative Jordan de- composition of g = ehu. The element s in the Weyl group of (G, A) is determined by yv ∈ G (not unique in general) in such a way that − −1 − yv ∈ N msMAN, where yhy = b(g) and G = ∪s∈W N msMAN is the Bruhat decomposition of G. 1. Introduction Given X ∈ GLn(C), the well-known QR decomposition asserts that X = QR, where Q is unitary and R is upper triangular with positive diagonal entries. The decomposition is unique. Let a(X) := diag R. Very recently it is known that [2] given A, B ∈ GLn(C), the following limit lim [a(AXmB)]1/m m→∞ exists and the limit is related to the eigenvalue moduli of X. More precisely, −1 Theorem 1.1. [2] Let A, B, X ∈ GLn(C). Let X = Y JY be the Jordan decomposition of X, where J is the Jordan form of X, diag J = diag (λ1, . , λn) satisfying |λ1| ≥ · · · ≥ |λn|. Then m 1/m lim [a(AX B)] = diag (|λω(1)|,..., |λω(n)|), m→∞ where the permutation ω is uniquely determined by the LωU decomposition of YB = LωU, where L is lower triangular and U is unit upper triangular.
    [Show full text]
  • Notes on Ergodic Theory
    Notes on ergodic theory Michael Hochman1 January 27, 2013 1Please report any errors to [email protected] Contents 1 Introduction 4 2 Measure preserving transformations 6 2.1 Measure preserving transformations . 6 2.2 Recurrence . 9 2.3 Inducedactiononfunctionsandmeasures . 11 2.4 Dynamics on metric spaces . 13 2.5 Some technicalities . 15 3 Ergodicity 17 3.1 Ergodicity . 17 3.2 Mixing . 19 3.3 Kac’s return time formula . 20 3.4 Ergodic measures as extreme points . 21 3.5 Ergodic decomposition I . 23 3.6 Measure integration . 24 3.7 Measure disintegration . 25 3.8 Ergodic decomposition II . 28 4 The ergodic theorem 31 4.1 Preliminaries . 31 4.2 Mean ergodic theorem . 32 4.3 Thepointwiseergodictheorem . 34 4.4 Generic points . 38 4.5 Unique ergodicity and circle rotations . 41 4.6 Sub-additive ergodic theorem . 43 5 Some categorical constructions 48 5.1 Isomorphismandfactors. 48 5.2 Product systems . 52 5.3 Natural extension . 53 5.4 Inverselimits ............................. 53 5.5 Skew products . 54 1 CONTENTS 2 6 Weak mixing 56 6.1 Weak mixing . 56 6.2 Weak mixing as a multiplier property . 59 6.3 Isometricfactors ........................... 61 6.4 Eigenfunctions . 62 6.5 Spectral isomorphism and the Kronecker factor . 66 6.6 Spectral methods . 68 7 Disjointness and a taste of entropy theory 73 7.1 Joinings and disjointness . 73 7.2 Spectrum, disjointness and the Wiener-Wintner theorem . 75 7.3 Shannon entropy: a quick introduction . 77 7.4 Digression: applications of entropy . 81 7.5 Entropy of a stationary process .
    [Show full text]
  • Arxiv:1409.2662V3 [Math.FA] 12 Nov 2014
    Measures and all that — A Tutorial Ernst-Erich Doberkat Math ++ Software, Bochum [email protected] November 13, 2014 arXiv:1409.2662v3 [math.FA] 12 Nov 2014 Abstract This tutorial gives an overview of some of the basic techniques of measure theory. It in- cludes a study of Borel sets and their generators for Polish and for analytic spaces, the weak topology on the space of all finite positive measures including its metrics, as well as mea- surable selections. Integration is covered, and product measures are introduced, both for finite and for arbitrary factors, with an application to projective systems. Finally, the duals of the Lp-spaces are discussed, together with the Radon-Nikodym Theorem and the Riesz Representation Theorem. Case studies include applications to stochastic Kripke models, to bisimulations, and to quotients for transition kernels. Page 1 EED. Measures Contents 1 Overview 2 2 Measurable Sets and Functions 3 2.1 MeasurableSets .................................. 4 2.1.1 A σ-AlgebraOnSpacesOfMeasures . 10 2.1.2 The Alexandrov Topology On Spaces of Measures . 11 2.2 Real-ValuedFunctions . 17 2.2.1 Essentially Bounded Functions . 21 2.2.2 Convergence almost everywhere and in measure ............. 22 2.3 Countably Generated σ-Algebras ......................... 27 2.3.1 Borel Sets in Polish and Analytic Spaces . ..... 32 2.3.2 Manipulating Polish Topologies . 38 2.4 AnalyticSetsandSpaces . .. .. .. .. .. .. .. .. 41 2.5 TheSouslinOperation ............................. 51 2.6 UniversallyMeasurableSets . ..... 55 2.6.1 Lubin’s Extension through von Neumann’s Selectors . ........ 58 2.6.2 Completing a Transition Kernel . 61 2.7 MeasurableSelections . 64 2.8 Integration ..................................... 67 2.8.1 FromMeasuretoIntegral .
    [Show full text]
  • Quotients of the Crown Domain by a Proper Action of a Cyclic Group
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 6, June 2014, Pages 3227–3239 S 0002-9947(2014)06006-6 Article electronically published on February 6, 2014 QUOTIENTS OF THE CROWN DOMAIN BY A PROPER ACTION OF A CYCLIC GROUP SARA VITALI Abstract. Let G/K be an irreducible Riemannian symmetric space of the non-compact type and denote by Ξ the associated crown domain. We show that for any proper action of a cyclic group Γ the quotient Ξ/ΓisStein.An analogous statement holds true for discrete nilpotent subgroups of a maximal split-solvable subgroup of G. We also show that Ξ is taut. Introduction Let G/K be an irreducible Riemannian symmetric space of the non-compact type, where G is assumed to be embedded in its universal complexification GC.In [AkGi90] D. N. Akhiezer and S. G. Gindikin pointed out a distinguished invari- ant domain Ξ of GC/KC containing G/K (as a maximal totally-real submanifold) such that the extended (left) G-action on Ξ is proper. The domain Ξ, which is usually referred to as the crown domain or the Akhiezer-Gindikin domain, is Stein and Kobayashi hyperbolic by a result of D. Burns, S. Hind and S. Halverscheid ([BHH03]; cf. [Bar03], [KrSt05]). In fact G. Fels and A.T. Huckleberry ([FeHu05]) have shown that with respect to these properties it is the maximal G-invariant com- plexification of G/K in GC/KC. By using the characterization of the G-invariant, plurisubharmonic functions on Ξ given in [BHH03], here we also note that Ξ is taut (Proposition 3.4).
    [Show full text]