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Ergodic Theory — an Operator- Theoretic Approach

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Ergodic Theory — an Operator- Theoretic Approach B. Farkas, T. Eisner, M. Haase, R. Nagel Ergodic Theory — An Operator- theoretic Approach – 12th International Internet Seminar – February 26, 2009 c by T. Eisner, B. Farkas, M. Haase, R. Nagel Contents 1 What is ergodic theory? ......................................... 1 2 Topological Dynamical Systems .................................. 7 2.1 Examples . 8 2.2 Basic constructions . 10 2.3 Topological Transitivity . 12 Exercises . 16 3 Minimality and Recurrence ..................................... 17 3.1 Minimality . 18 3.2 Topological Recurrence . 20 3.3 Recurrence in Extensions . 23 Exercises . 25 4 The Space C(K) and the Induced Operator ........................ 27 4.1 The space C(K) as a commutative C∗-algebra . 28 4.2 The Theorem of Gelfand–Naimark . 31 4.3 The Induced Operator . 33 Exercises . 35 5 Measure-Preserving Systems .................................... 37 5.1 Examples . 39 5.2 Measures on Compact Spaces . 42 5.3 Haar Measure and Rotations . 44 Exercises . 45 6 Recurrence and Ergodicity ...................................... 47 6.1 The Measure Algebra and Invertible MDS . 47 6.2 Recurrence . 49 6.3 Ergodicity . 53 Exercises . 57 v vi Contents 7 The Banach Lattice Lp and the Induced Operator . 59 7.1 The Space Lp(Ω,Σ, µ) as a Banach Lattice . 60 7.2 The Induced Operator and Ergodicity . 65 7.3 Isomorphism of Measure Preserving Systems . 67 Exercises . 69 8 The Mean Ergodic Theorem I ................................... 71 8.1 Von Neumann’s Mean Ergodic Theorem . 72 8.2 Mean Ergodic Operators on Banach Spaces . 75 8.3 Examples . 78 Supplement: Powers and Convex Combinations of Mean Ergodic Operators . 79 Exercises . 81 9 The Mean Ergodic Theorem II .................................. 83 9.1 More on Ergodicity . 83 9.2 From Topological to Measure-Preserving Systems . 87 9.3 Application: Equidistribution . 91 Exercises . 94 10 The Pointwise Ergodic Theorem ................................. 97 10.1 Pointwise Ergodic Operators . 98 10.2 Banach’s Principle and Maximal Inequalities . 100 10.3 Applications . 103 Exercises . 108 11 Compact Semigroups and Groups ................................111 11.1 Compact Semigroups . 112 11.2 Compact Groups . 115 11.3 The Character Group. 116 11.4 The Pontryagin Duality Theorem . 119 11.5 Application: Ergodic Rotations and Kronecker’s Theorem . 121 Exercises . 127 12 The Jacobs–de Leeuw–Glicksberg Decomposition . 129 12.1 Compact Group Actions on Banach Spaces . 133 12.2 Almost Weakly Stable Vectors. 135 13 Dynamical Systems with Discrete Spectrum . 143 13.1 Monothetic Groups . 144 13.2 Minimal TDSs with Discrete Spectrum . 146 13.3 Ergodic MDSs with Discrete Spectrum . 148 13.4 Examples . 150 Exercises . 152 Contents vii 14 Mixing Dynamical Systems ......................................155 14.1 Strong mixing . 156 14.2 Weak mixing . 159 14.3 Weak mixing of all orders . 163 Exercises . 167 15 A Glimpse on Arithmetic Progressions............................169 15.1 The Furstenberg Correspondence Principle . 171 15.2 A Multiple Ergodic Theorem . 173 15.3 The Furstenberg–Sark´ ozy¨ Theorem . 178 A Topology ......................................................181 B Measure and Integration Theory .................................191 C Functional Analysis ............................................209 References .........................................................223 Lecture 1 What is ergodic theory? Ergodic Theory is a recent mathematical discipline and its name, in contrast to, e.g., number theory, does not explain its subject. However, its origin can be described quite precisely. It was around 1880 when L. Boltzmann, J. C. Maxwell and others tried to explain thermodynamical phenomena by mechanical models and their underlying mathe- matical principles. In this context, L. Boltzmann [Boltzmann (1885)] coined the word Ergode (as a special kind of Monode)1 “Monoden, welche nur durch die Gleichung der lebendigen Kraft beschrankt¨ sind, will ich als Ergoden bezeichnen.” A few years later (in 1911) P. and T. Ehrenfest [Ehrenfest (1912)] wrote “... haben Boltzmann und Maxwell eine Klasse von mechanischen Systemen durch die fol- gende Forderung definiert: Die einzelne ungestorte¨ Bewegung des Systems fuhrt¨ bei unbegrenzter Fortsetzung schließlich durch jeden Phasenpunkt hindurch, der mit der mitgegebenen Totalenergie vertraglich¨ ist. – Ein mechanisches System, das diese Forderung erfullt,¨ nennt Boltzmann ein ergodisches System.” The assumption that certain systems are “ergodic” is then called “Ergodic Hypothe- sis”. Leaving the original problem behind, “Ergodic Theory” set out on its fascinat- ing journey into mathematics and arrived at quite unexpected destinations. Before we, too, undertake this journey, let us explain the original problem without going too deep into the underlying physics. We start with an (ideal) gas contained in a box and represented by d (frictionless) moving particles. Each particle is described by six coordinates (three for position, three for velocity), so the situation of the gas d (better: the state of the system) is given by a point ω ∈ R6 . Clearly, not all points d in R6 can be attained by our gas in the box, so we restrict our considerations to the set Ω of all possible states and call it the state space of our system. We now observe 1 For the still controversial discussion “On the origin of the notion ’Ergodic Theory”’ see the excellent article by M. Mathieu [Mathieu (1988)] but also the remarks by G. Gallavotti in [Gallavotti (1975)]. 1 2 1 What is ergodic theory? that our system changes while time is running, i.e., the particles are moving (in the box) and therefore a given state (= point in Ω) also “moves” (in Ω). This motion (in the box, therefore in Ω) is governed by Newton’s laws of mechanics and then by Hamilton’s differential equations. The solutions to these equations determine a map ϕ : Ω −→ Ω in the following way: If our system, at time t = 0, is in the state ω0 ∈ Ω, then at time t = 1 it will be in a new state ω1, and we define ϕ by ϕ(ω0) := ω1. As a consequence, at time t = 2 the state ω0 becomes 2 ω2 := ϕ(ω1) = ϕ (ω0) and n ωn := ϕ (ω0) n at time t = n ∈ N. We call {ϕ (ω0) | n ∈ N0} the orbit of ω0. In this way, the physical motion of the system of particles becomes a “motion” of the points in the state space. The motion of all states within one time unit is given by the map ϕ. For these objects we introduce the following terminology. Definition 1.1. A pair (Ω,ϕ) consisting of a state space Ω and a map ϕ : Ω −→ Ω is called a dynamical system. 2 The mathematical goal now is not so much to determine ϕ but rather to find inter- esting properties of ϕ. Motivated by the underlying physical situation, the emphasis is on “long term” properties of ϕ, i.e., properties of ϕn as n gets large. First objection. In the physical situation it is not possible to determine exactly the n given initial state ω0 ∈ Ω of the system or any of its later states ϕ (ω0). Fig. 1.1 Try to determine the exact state of the system for only n = 1000 gas molecules! To overcome this objection we introduce “observables”, i.e., functions f : Ω −→ R assigning to each state ω ∈ Ω the value f (ω) of a measurement, for instance of 2 This is a mathematical model for the philosophical principle of determinism and we refer to the article by G. Nickel in [Engel and Nagel (2000)] for more on this interplay between mathematics and philosophy. 1 What is ergodic theory? 3 the temperature. The motion in time (“evolution”) of the states described by the map ϕ : Ω −→ Ω is then reflected by a map Tϕ of the observables defined as f 7−→ Tϕ f := f ◦ ϕ. This change of perspective is not only physically justified, it also has an enormous mathematical advantage: The set of all observables { f : Ω −→ R} has a vector space structure and the map Tϕ = ( f 7−→ f ◦ϕ) becomes a linear operator on this vector space. n So instead of looking at the orbits ϕ (ω0) of the state map ϕ we study the orbit n {(Tϕ ) f | n ∈ N0} of an observable f under the linear operator Tϕ . This change of perspective allows the use of operator theoretic tools and is the basis of our ap- proach. Returning to our description of the motion of the particles in a box and keeping in mind realistic experiments, we should make another objection. Second objection. The motion of our system happens so quickly that we will not be able to determine the states 2 ω0,ϕ(ω0),ϕ (ω0),... nor their measurements 2 f (ω0),Tϕ f (ω0),Tϕ f (ω0),... at time t = 0,1,2,.... Reacting on this objection we slightly change.
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