DOCSLIB.ORG

Ergodic Theory in the Perspective of Functional Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ergodic Theory in the Perspective of Functional Analysis Ergodic Theory in the Perspective of Functional Analysis 13 Lectures by Roland Derndinger, Rainer Nagel, GÄunther Palm (uncompleted version) In July 1984 this manuscript has been accepted for publication in the series \Lecture Notes in Mathematics" by Springer-Verlag. Due to a combination of unfortunate circumstances none of the authors was able to perform the necessary ¯nal revision of the manuscript. TÄubingen,July 1987 1 2 I. What is Ergodic Theory? The notion \ergodic" is an arti¯cial creation, and the newcomer to \ergodic theory" will have no intuitive understanding of its content: \elementary ergodic theory" neither is part of high school- or college- mathematics (as does \algebra") nor does its name explain its subject (as does \number theory"). Therefore it might be useful ¯rst to explain the name and the subject of \ergodic theory". Let us begin with the quotation of the ¯rst sentence of P. Walters' introductory lectures (1975, p. 1): \Generally speaking, ergodic theory is the study of transformations and flows from the point of view of recurrence properties, mixing properties, and other global, dynamical, properties connected with asymptotic behavior." Certainly, this de¯nition is very systematic and complete (compare the beginning of our Lectures III. and IV.). Still we will try to add a few more answers to the question: \What is Ergodic Theory ?" Naive answer: A container is divided into two parts with one part empty and the other ¯lled with gas. Ergodic theory predicts what happens in the long run after we remove the dividing wall. First etymological answer: ergodhc=di±cult. Historical answer: 1880 - Boltzmann, Maxwell - ergodic hypothesis 1900 - Poincar¶e - recurrence theorem 1931 - von Neumann - mean ergodic theorem 1931 - Birkho® - individual ergodic theorem 1958 - Kolmogorov - entropy as an invariant 1963 - Sinai - billiard flow is ergodic 1970 - Ornstein - entropy classi¯es Bernoulli shifts 1975 - Akcoglu - individual Lp-ergodic theorem Naive answer of a physicist: Ergodic theory proves that time mean equals space mean. I.E. Farquhar's [1964] answer: \Ergodic theory originated as an o®shot of the work of Boltzmann and of Maxwell in the kinetic theory of gases. The impetus provided by the physical problem led later to the development by pure mathemati- cians of ergodic theory as a branch of measure theory, and, as is to be expected, the scope of this mathematical theory extends now far beyond the initial ¯eld of interest. However, the chief physical problems to which ergodic theory has rel- evance, namely, the justi¯cation of the methods of statistical mechanics and the relation between reversibility and irreversibility have been by no means satisfacto- rily solved, and the question arises of how far the mathematical theory contributes to the elucidation of these physical problems." 3 Physicist's answer: Reality Physical model Mathematical consequences A gas with n particles The \state" of the gas is a at time t 0 is given. point x in the \state space" X R6n. Time changes Time change is described by Theorem of Liouville: the Hamiltonian di®erential ' preserves the (normal- equations. Their solutions ized) Lebesgue measure ¹ Ñ yield a mapping ' : X on X. X, such that the state x0 at time t 0 becomes the state x1 'px0q at time t 1. De¯nition: An observable is a function f : X Ñ R, where fpxq can be regarded the long run behavior as the outcome of a mea- is observed. surement, when the gas is in the state P X. Problem: Find lim fp'npxqq! 1st objection: Modi¯ed problem: Find Time change is much the time mean Mtfpxq : faster than our obser- ° 1 n1 p ip qq vations. lim n i0 f ' x ! 2nd objection: In Additional hypothesis \Theorem" 1: If the er- practice, it is impos- (ergodic hypothesis): Each godic hypothesis is satis- sible to determine the particular motion will pass ³¯ed, we have Mtfpxq state x. through every state consis- f d¹ space mean, which tent with its energy (see is independent of the state P.u.T. Ehrenfest 1911). x. \Theorem" 2: The er- godic hypothesis is \never" satis¯ed. Ergodic theory looks for better ergodic hypothesis and better \ergodic theorems". Commonly accepted etymological answer: êrgon = energy –ὁδός = {path (P. and T. Ehrenfest 1911, p. 30) 4 \Correct" etymological answer: êrgon = energy –ῶδης = {like (Boltzmann 1884/85, see also III.) K. Jacobs' [1965] answer: \... als EinfÄuhrungfÄursolche Leser gedacht, die gern einmal erfahren mÄochten, womit sich diese Theorie mit dem seltsamen, aus den griechischen WÄortern er- gon (Arbeit) und odoc (Weg) zusammengesetzten Namen eigentlich beschÄaftigt. Die Probleme der Ergodentheorie kreisen um einen Begri®, der einerseits so viele reizvolle SpezialfÄalleumfa¼t, da¼ sowohl der Polyhistor als auch der stille Genie¼er auf ihre Kosten kommen, andererseits so einfach ist, da¼ sich die zentralen Ergeb- nisse und Probleme der Ergodentheorie leicht darstellen lassen; diese einfach zu formulierenden Fragestellungen erfordern jedoch bei naherer Untersuchung oft de- rartige Anstrengungen, da¼ harte Arbeiter hier ihr rechtes VergnÄugen¯nden wer- den." J. Dieudonne's [1977] answer: \Le point de d¶epartde la th¶eorieergodique provient du d¶eveloppement de la m¶ecaniquestatistique et de la theorie cin¶etiquedes gaz, o`ul'exp¶eriencesugg`ere und tendence `al'\uniformite": si l'on consid`ere`aun instant donn¶eun m¶elange h¶et¶erog`enede plusieurs gaz, l'¶evolution du m¶elangeau cours du temps tend `ale rendre homog¶ene." W. Parry's [1981] answer: \Ergodic Theory is di±cult to characterize, as it stands at the junction of so many areas, drawing on the techniques and examples of probability theory, vector ¯elds on manifolds, group actions on homogeneous spaces, number theory, statistical mechanics, etc..."" (e.g. functional analysis; added by the authors). Elementary mathematical answer: Let X be a set, ' : X Ñ X a mapping. The induced operator T' maps functions f : X Ñ R into T'f : f '. Ergodic theory investigates the asymptotic behavior n n P of ' and T' for n N. Our answer: More structure is needed on the set X, usually at least a topological or a measure theoretical structure. In both cases we can study the asymptotic behavior of the n powers T of the linear operator T T', de¯ned either on the Banach space CpXq of all continuous functions on X or on the Banach space L1pX; §; ¹q of all ¹-integrable functions on X. 5 II. Dynamical Systems Many of the answers presented in Lecture I indicate that ergodic theory deals with pairs pX; 'q where X is a set whose points represent the \states" of a physical system while ' is a mapping from X into X describing the change of states after one time unit. The ¯rst step towards a mathematical theory consists in ¯nding out which abstract properties of the physical state spaces will be essential. It is evident that an \ergodic theory" based only on set-theoretical assumptions is of little interest. Therefore we present three di®erent mathematical structures which can be imposed on the state space X and the mapping ' in order to yield \dynamical systems" that are interesting from the mathematical point of view. The parallel development of the corresponding three \ergodic theories" and the investigation of their mutual interaction will be one of the characteristics of the following lectures. II.1 De¯nition: (i) pX; §; ¹; 'q is a measure-theoretical dynamical system (briefly: MDS) if pX; §; ¹q is a probability space and ' : X Ñ X is a bi-measure-preserving transforma- tion. (ii) pX; 'q is a topological dynamical system (TDS) if X is a compact space and ' : X Ñ X is homeomorphism. (iii) pE; T q is a functional-analytic dynamical system (FDS) if E is a Banach space and T : E Ñ E is a bounded linear operator. Remarks: 1. The term \bi-measure-preserving" for the transformation ' : X Ñ X in (i) is to be understood in the following sense: There exists a subset X0 of X with ¹pX0q 1 such that the restriction '0 : X0 Ñ X0 of ' is bijective, and both '0 and its inverse are measurable and measure-preserving for the induced σ-algebra §0 : tA X X0 : A P §u. 2. If ' is bi-measure-preserving with respect to ¹, we call ¹ a '-invariant measure. 3. As we shall see in (II.4) every MDS and TDS leads to an FDS in a canonical way. Thus a theory of FDSs can be regarded as a joint generalization of the topological theory of TDSs and the probabilistic theory of MDSs. In most of the following chapters we will either start from or aim for a formulation of the main theorem(s) in the language of FDSs. 4. DDSs (\di®erentiable dynamical systems") will not be investigated in these lec- tures (see Bowen [1975], Smale [1967], [1980]). Before proving any results we present in this lecture the fundamental (types of) examples of dynamical systems which will frequently reappear in the ensuing text. The reader is invited to apply systematically every de¯nition and result to at least some of these examples. II.2. Rotations: (i) Let ¡ tz P C : |z| 1u be the unit circle, § its Borel algebra, and m the normalized Lebesgue measure on ¡. Choose an a P ¡ and de¯ne 'apzq : a z for all z P ¡: Clearly, p¡; 'aq is a TDS, and p¡; B; m; 'aq an MDS. 6 (ii) A more abstract version of the above example is the following: Take a compact group G with Borel algebra B and normalized Haar measure m. Choose h P G and de¯ne the (left)rotation 'hpgq : h g for all g P G: Again, pG; 'hq is a TDS, and pG; B; m; 'hq an MDS.
Load more

Recommended publications
  • Symmetry and Ergodicity Breaking, Mean-Field Study of the Ising Model
    Symmetry and ergodicity breaking, mean-field study of the Ising model Rafael Mottl, ETH Zurich Advisor: Dr. Ingo Kirsch Topics Introduction Formalism The model system: Ising model Solutions in one and more dimensions Symmetry and ergodicity breaking Conclusion Introduction phase A phase B Tc temperature T phase transition order parameter Critical exponents Tc T Formalism Statistical mechanical basics sample region Ω i) certain dimension d ii) volume V (Ω) iii) number of particles/sites N(Ω) iv) boundary conditions Hamiltonian defined on the sample region H − Ω = K Θ k T n n B n Depending on the degrees of freedom Coupling constants Partition function 1 −βHΩ({Kn},{Θn}) β = ZΩ[{Kn}] = T r exp kBT Free energy FΩ[{Kn}] = FΩ[K] = −kBT logZΩ[{Kn}] Free energy per site FΩ[K] fb[K] = lim N(Ω)→∞ N(Ω) i) Non-trivial existence of limit ii) Independent of Ω iii) N(Ω) lim = const N(Ω)→∞ V (Ω) Phases and phase boundaries Supp.: -fb[K] exists - there exist D coulping constants: {K1,...,KD} -fb[K] is analytic almost everywhere - non-analyticities of f b [ { K n } ] are points, lines, planes, hyperplanes in the phase diagram Dimension of these singular loci: Ds = 0, 1, 2,... Codimension C for each type of singular loci: C = D − Ds Phase : region of analyticity of fb[K] Phase boundaries : loci of codimension C = 1 Types of phase transitions fb[K] is everywhere continuous. Two types of phase transitions: ∂fb[K] a) Discontinuity across the phase boundary of ∂Ki first-order phase transition b) All derivatives of the free energy per site are continuous
    [Show full text]
  • Functional Analysis Lecture Notes Chapter 2. Operators on Hilbert Spaces
    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 2. OPERATORS ON HILBERT SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples First recall the basic definitions regarding operators. Definition 1.1 (Continuous and Bounded Operators). Let X, Y be normed linear spaces, and let L: X Y be a linear operator. ! (a) L is continuous at a point f X if f f in X implies Lf Lf in Y . 2 n ! n ! (b) L is continuous if it is continuous at every point, i.e., if fn f in X implies Lfn Lf in Y for every f. ! ! (c) L is bounded if there exists a finite K 0 such that ≥ f X; Lf K f : 8 2 k k ≤ k k Note that Lf is the norm of Lf in Y , while f is the norm of f in X. k k k k (d) The operator norm of L is L = sup Lf : k k kfk=1 k k (e) We let (X; Y ) denote the set of all bounded linear operators mapping X into Y , i.e., B (X; Y ) = L: X Y : L is bounded and linear : B f ! g If X = Y = X then we write (X) = (X; X). B B (f) If Y = F then we say that L is a functional. The set of all bounded linear functionals on X is the dual space of X, and is denoted X0 = (X; F) = L: X F : L is bounded and linear : B f ! g We saw in Chapter 1 that, for a linear operator, boundedness and continuity are equivalent.
    [Show full text]
  • 18.102 Introduction to Functional Analysis Spring 2009
    MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 108 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 19. Thursday, April 16 I am heading towards the spectral theory of self-adjoint compact operators. This is rather similar to the spectral theory of self-adjoint matrices and has many useful applications. There is a very effective spectral theory of general bounded but self- adjoint operators but I do not expect to have time to do this. There is also a pretty satisfactory spectral theory of non-selfadjoint compact operators, which it is more likely I will get to. There is no satisfactory spectral theory for general non-compact and non-self-adjoint operators as you can easily see from examples (such as the shift operator). In some sense compact operators are ‘small’ and rather like finite rank operators. If you accept this, then you will want to say that an operator such as (19.1) Id −K; K 2 K(H) is ‘big’. We are quite interested in this operator because of spectral theory. To say that λ 2 C is an eigenvalue of K is to say that there is a non-trivial solution of (19.2) Ku − λu = 0 where non-trivial means other than than the solution u = 0 which always exists. If λ =6 0 we can divide by λ and we are looking for solutions of −1 (19.3) (Id −λ K)u = 0 −1 which is just (19.1) for another compact operator, namely λ K: What are properties of Id −K which migh show it to be ‘big? Here are three: Proposition 26.
    [Show full text]
  • 5 Toeplitz Operators
    2008.10.07.08 5 Toeplitz Operators There are two signal spaces which will be important for us. • Semi-infinite signals: Functions x ∈ ℓ2(Z+, R). They have a Fourier transform g = F x, where g ∈ H2; that is, g : D → C is analytic on the open unit disk, so it has no poles there. • Bi-infinite signals: Functions x ∈ ℓ2(Z, R). They have a Fourier transform g = F x, where g ∈ L2(T). Then g : T → C, and g may have poles both inside and outside the disk. 5.1 Causality and Time-invariance Suppose G is a bounded linear map G : ℓ2(Z) → ℓ2(Z) given by yi = Gijuj j∈Z X where Gij are the coefficients in its matrix representation. The map G is called time- invariant or shift-invariant if it is Toeplitz , which means Gi−1,j = Gi,j+1 that is G is constant along diagonals from top-left to bottom right. Such matrices are convolution operators, because they have the form ... a0 a−1 a−2 a1 a0 a−1 a−2 G = a2 a1 a0 a−1 a2 a1 a0 ... Here the box indicates the 0, 0 element, since the matrix is indexed from −∞ to ∞. With this matrix, we have y = Gu if and only if yi = ai−juj k∈Z X We say G is causal if the matrix G is lower triangular. For example, the matrix ... a0 a1 a0 G = a2 a1 a0 a3 a2 a1 a0 ... 1 5 Toeplitz Operators 2008.10.07.08 is both causal and time-invariant.
    [Show full text]
  • 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]
  • Mite Beometrles Ana Corn Binatorial Designs
    AMERICAN MATHEMATICAL SOCIETY mite beometrles ana Corn binatorial Designs Proceedings of the AMS Special Session in Finite Geometries and Combinatorial Designs held October 29-November 1, 1987 http://dx.doi.org/10.1090/conm/111 Titles in This Series Volume 1 Markov random fields and their 19 Proceedings of the Northwestern applications, Ross Kindermann and homotopy theory conference, Haynes J. Laurie Snell R. Miller and Stewart B. Priddy, Editors 2 Proceedings of the conference on 20 Low dimensional topology, Samuel J. integration, topology, and geometry in Lomonaco, Jr., Editor linear spaces, William H. Graves, Editor 21 Topological methods in nonlinear 3 The closed graph and P-closed functional analysis, S. P. Singh, graph properties in general topology, S. Thomaier, and B. Watson, Editors T. R. Hamlett and L. L. Herrington 22 Factorizations of b" ± 1, b = 2, 4 Problems of elastic stability and 3, 5, 6, 7, 10, 11, 12 up to high vibrations, Vadim Komkov, Editor powers, John Brillhart, D. H. Lehmer, 5 Rational constructions of modules J. L. Selfridge, Bryant Tuckerman, and for simple Lie algebras, George B. S. S. Wagstaff, Jr. Seligman 23 Chapter 9 of Ramanujan's second 6 Umbral calculus and Hopf algebras, notebook-Infinite series identities, Robert Morris, Editor transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 7 Complex contour integral representation of cardinal spline 24 Central extensions, Galois groups, and functions, Walter Schempp ideal class groups of number fields, A. Frohlich 8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio, 25 Value distribution theory and its Editors applications, Chung-Chun Yang, Editor 9 Papers in algebra, analysis and 26 Conference in modern analysis statistics, R.
    [Show full text]
  • L∞ , Let T : L ∞ → L∞ Be Defined by Tx = ( X(1), X(2) 2 , X(3) 3 ,... ) Pr
    ASSIGNMENT II MTL 411 FUNCTIONAL ANALYSIS 1. For x = (x(1); x(2);::: ) 2 l1, let T : l1 ! l1 be defined by ( ) x(2) x(3) T x = x(1); ; ;::: 2 3 Prove that (i) T is a bounded linear operator (ii) T is injective (iii) Range of T is not a closed subspace of l1. 2. If T : X ! Y is a linear operator such that there exists c > 0 and 0 =6 x0 2 X satisfying kT xk ≤ ckxk 8 x 2 X; kT x0k = ckx0k; then show that T 2 B(X; Y ) and kT k = c. 3. For x = (x(1); x(2);::: ) 2 l2, consider the right shift operator S : l2 ! l2 defined by Sx = (0; x(1); x(2);::: ) and the left shift operator T : l2 ! l2 defined by T x = (x(2); x(3);::: ) Prove that (i) S is a abounded linear operator and kSk = 1. (ii) S is injective. (iii) S is, in fact, an isometry. (iv) S is not surjective. (v) T is a bounded linear operator and kT k = 1 (vi) T is not injective. (vii) T is not an isometry. (viii) T is surjective. (ix) TS = I and ST =6 I. That is, neither S nor T is invertible, however, S has a left inverse and T has a right inverse. Note that item (ix) illustrates the fact that the Banach algebra B(X) is not in general commutative. 4. Show with an example that for T 2 B(X; Y ) and S 2 B(Y; Z), the equality in the submulti- plicativity of the norms kS ◦ T k ≤ kSkkT k may not hold.
    [Show full text]
  • Josiah Willard Gibbs
    GENERAL ARTICLE Josiah Willard Gibbs V Kumaran The foundations of classical thermodynamics, as taught in V Kumaran is a professor textbooks today, were laid down in nearly complete form by of chemical engineering at the Indian Institute of Josiah Willard Gibbs more than a century ago. This article Science, Bangalore. His presentsaportraitofGibbs,aquietandmodestmanwhowas research interests include responsible for some of the most important advances in the statistical mechanics and history of science. fluid mechanics. Thermodynamics, the science of the interconversion of heat and work, originated from the necessity of designing efficient engines in the late 18th and early 19th centuries. Engines are machines that convert heat energy obtained by combustion of coal, wood or other types of fuel into useful work for running trains, ships, etc. The efficiency of an engine is determined by the amount of useful work obtained for a given amount of heat input. There are two laws related to the efficiency of an engine. The first law of thermodynamics states that heat and work are inter-convertible, and it is not possible to obtain more work than the amount of heat input into the machine. The formulation of this law can be traced back to the work of Leibniz, Dalton, Joule, Clausius, and a host of other scientists in the late 17th and early 18th century. The more subtle second law of thermodynamics states that it is not possible to convert all heat into work; all engines have to ‘waste’ some of the heat input by transferring it to a heat sink. The second law also established the minimum amount of heat that has to be wasted based on the absolute temperatures of the heat source and the heat sink.
    [Show full text]
  • James H. Olsen
    James H. Olsen Contact Department of Mathematics Information North Dakota State University Voice: (614) 292-2572 205 Dreese Labs Fax: (614) 292-7596 2015 Neil Avenue E-mail: [email protected] Columbus, OH 43210 USA WWW: math.ndsu.nodak.edu/faculty/olsen/ Research Ergodic theory Interests Education The University of Minnesota, Minneapolis, Minnesota USA Ph.D., Mathematics, 1968 • Dissertation Title: Dominated Estimates of Lp Contractions, 1 < p < ∞. • Advisor: Professor Rafael V. Chacon • Area of Study: Ergodic Theory M.S., Mathematics, 1967 • Advisor: Professor Rafael V. Chacon • Area of Study: Ergodic Theory N.C. State College, Raleigh, North Carolina USA B.S., Electrical Engineering, 1963 Awards and Eta Kappa Nu Honors • Member Scabbard and Blade National Honor Society • First Sergeant Academic and North Dakota State University, Fargo, ND USA Professional Experience Professor of Mathematics 1984 to present Associate Professor of Mathematics 1974 to 1984 Assistant Professor of Mathematics 1970 to 1974 Mathematical Research Division, National Security Agency, Fort George G. Meade, Maryland, USA Research Mathematician 1969 to 1970 • Active duty with the U.S. Army. • Achieved the rank of captain. The University of Minnesota, Minneapolis, Minnesota USA Instructor, School of Mathematics Fall 1968 Nuclear Effects Lab, White Sands Missle Range, New Mexico USA Electronics Engineer Summer 1964 1 of 5 Publications Dominated estimates of positive contractions, (with R.V. Chacon), Proceedings of the American Mathematical Society, 20, 226-271 (1969). Dominated estimates of convex combinations of commuting isometries, Israel Journal of Mathematics, 11, 1-13 (1972). Estimates of convex combinations of commuting isometries, Z. Wahrscheinlicheits- theorie verw. Geb. 26, 317-324 (1973).
    [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]
  • Operators, Functions, and Systems: an Easy Reading
    http://dx.doi.org/10.1090/surv/092 Selected Titles in This Series 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmiiller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W.
    [Show full text]
  • Viscosity from Newton to Modern Non-Equilibrium Statistical Mechanics
    Viscosity from Newton to Modern Non-equilibrium Statistical Mechanics S´ebastien Viscardy Belgian Institute for Space Aeronomy, 3, Avenue Circulaire, B-1180 Brussels, Belgium Abstract In the second half of the 19th century, the kinetic theory of gases has probably raised one of the most impassioned de- bates in the history of science. The so-called reversibility paradox around which intense polemics occurred reveals the apparent incompatibility between the microscopic and macroscopic levels. While classical mechanics describes the motionof bodies such as atoms and moleculesby means of time reversible equations, thermodynamics emphasizes the irreversible character of macroscopic phenomena such as viscosity. Aiming at reconciling both levels of description, Boltzmann proposed a probabilistic explanation. Nevertheless, such an interpretation has not totally convinced gen- erations of physicists, so that this question has constantly animated the scientific community since his seminal work. In this context, an important breakthrough in dynamical systems theory has shown that the hypothesis of microscopic chaos played a key role and provided a dynamical interpretation of the emergence of irreversibility. Using viscosity as a leading concept, we sketch the historical development of the concepts related to this fundamental issue up to recent advances. Following the analysis of the Liouville equation introducing the concept of Pollicott-Ruelle resonances, two successful approaches — the escape-rate formalism and the hydrodynamic-mode method — establish remarkable relationships between transport processes and chaotic properties of the underlying Hamiltonian dynamics. Keywords: statistical mechanics, viscosity, reversibility paradox, chaos, dynamical systems theory Contents 1 Introduction 2 2 Irreversibility 3 2.1 Mechanics. Energyconservationand reversibility . ........................ 3 2.2 Thermodynamics.
    [Show full text]