DOCSLIB.ORG

The Pointwise Ergodic Theorem and Its Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Pointwise Ergodic Theorem and Its Applications The Pointwise Ergodic Theorem and its Applications Peter Oberly 11/9/2018 Introduction Algebra has homomorphisms and topology has continuous maps; in these notes we explore the structure preserving maps for measure theory known (somewhat unimaginatively) as measure preserving transformations. The first section contains some (but not all) of the necessary definitions for this talk and in the second we introduce some classical examples to illustrate these definitions. We then turn our attention to the dynamics of measure preserving maps which leads us to the pointwise ergodic theorem. In the final section we use the ergodic theorem to prove Borel's theorem on normal numbers. Definitions Definition. A σ-algebra A is a collection of subsets of a non-empty set X so that X 2 A and A is closed under complementation and countable unions; The pair (X; A) is called a measurable space, and elements of A are called measurable sets. A particularly important σ-algebra is the collection of Borel sets, defined to be the σ-algebra generated by the open subsets of a topological space X. Definition. A measure m : A! [0; 1] is a function which satisfies the following: 1. m(E) ≥ 0 for all E 2 A; 2. m(;) = 0; 1 S P 3. If fEngn=1 ⊂ A is a sequence of pairwise disjoint sets in A, then m( n En) = n m(En). A measure space is a triple (X; A; m) where (X; A) is a measurable space and m is a measure defined on A. The triple (X; A; m) is called a probability space if m(X) = 1. Definition. Let (X; A; m) and (Y; B; n) be measure spaces, and let T : X ! Y be a map from X into Y . T is said to be measurable if T −1(E) 2 A for each E 2 B; that is, if the pre-image of every measurable set is measurable. Definition. A measurable transformation T :(X; A; m) ! (Y; B; n) is said to be measure- preserving if m(T −1(E)) = n(E) for all E 2 B. If T is a bijection and T −1 is also measure preserving, then T is said to be invertible. If (X; A; m) is a probability space, and if T : X ! X is measure preserving, then the quadruple (X; A; m; T ) is sometimes referred to as a measurable dynamical system. Remarks: (1) We should really write T :(X; A; m) ! (Y; B; n) since the measure preserving property Notes on the Ergodic Theorem depends on both the σ-algebras and the measures, but will often write T : X ! Y instead. (2) If T :(X; A; ; m) ! (Y; B; n) and S :(Y; B; n) ! (Z; C; p) are measure preserving, then so is S ◦ T . (3) Measure preserving maps are the structure preserving transformations (morphisms) of measure spaces. (4) As such, a measure preserving map T : X ! X induces a morphism on the Banach 1 1 1 space of m-integrable functions L (m). In detail, let UT : L (m) !L (m) be defined by UT (f) = f ◦ T . It is evident that UT is linear, and if f ≥ 0 (and so is real valued), then (UT f)(x) = f(T (x)) ≥ 0 for x 2 X. So UT is positive. In fact, UT is an isometry. For if Pn s is a non-negative simple function s = k=1 akχAk , where ak are scalars and Ak are the measurable sets where s > 0, then Z n Z n n Z X X −1 X UT (s) dm = ak χAk ◦ T dm = akm(T (Ak)) = akm(Ak) = s dm: k=1 k=1 k=1 Therefore choosing a sequence of simple functions sn which converges monotonically to jfj, 1 where f 2 L (m), shows jjUT (f)jj1 = jjfjj1. Note also that this shows UT really does map into L1(m). (5) As we are interested in the dynamics of measure preserving maps, from now on we will restrict our attention to measurable functions T : X ! X. Additionally, unless other wise stated, we will assume that (X; A; m) is a probability space. Our last definition requires a bit of motivation. Let (X; A; m; T ) be a measurable dynamical system. If T −1(E) = E for E 2 A, then T −1(X n E) = X n E and we could study our system by examining the two simpler systems (E; A\E; mjE A\E;T jE) and (X n E; A\(X n E); mjA\(XnE);T jXnE) (with the corresponding measures normalized appro- priately). If 0 < m(E) < 1, then we have actually decomposed our original system into two smaller ones. However, if m(E) = 0 or m(X n E) = 0 (i.e. m(E) = 1), then one of our simpler systems is in fact trivial, and we are left with a system essentially the same as the one we started with. It follows that those measurable dynamical systems where T −1(E) = E implies m(E) = 0 or 1 are not usefully decomposable in this way. It makes sense therefore to study those systems where such decomposition is not possible, for understanding these will enable us to understand the ones which can be simplified. We call such systems ergodic. Definition. A measurable dynamical system (X; A; m; T ) is said to be ergodic if E 2 A and T −1(E) = E implies that m(E) = 0 or 1. We will often have a specific probability space (X; A; m) in mind and refer to the measure preserving transformation T as ergodic. There are many characterizations of ergodicity; one which will prove useful in this talk is the following. Theorem 1. (X; A; m; T ) is ergodic if and only if f 2 L1(m) and f ◦ T = f ae implies that f is constant ae. Proof. Assume that for all f 2 L1(m) that if f ◦ T = f ae then f is constant ae. Let E 2 A −1 1 be so that T (E) = E. Then χE ◦ T = χE. As χE 2 L (m), then χE is constant ae. Therefore χE is either 0 or 1 ae and so m(E) = 0 or 1. The converse is more technical, and can be found in McDonald and Wiess on page 616. 2 Notes on the Ergodic Theorem Examples n n (1) Let T : R ! R be a linear map and let m be the Lebesgue measure on the Borel sets n n of R . If T is singular, then range T is a proper subspace of R , and so T is not measure pre- serving. If instead T is non-singular, from linear algebra then m(T −1(E)) = m(E)=j det T j for all Borel sets E. Therefore T is a measure preserving linear map if and only if j det T j = 1. (2) Let S1 = fz 2 C : jzj = 1g and let B denote the Borel σ-algebra. Then with normalized circular Lebesgue measure m, the triple (S1; B; m) is a probability space. For a 2 S1, define 1 1 the rotation Ta : S ! S by Ta(z) = az. Then Ta is measure preserving and invertible for all a. It is very instructive to show the following. Theorem 2. The rotation T = Ta is ergodic if and only if a is not a root of unity Proof. Suppose that a is a root of unity. Then ap = 1 for some p 6= 0. Let f : S1 ! S1 be defined by f(z) = zp. Then (f ◦ T )(z) = f(az) = apzp = f(z) for all z 2 S1. Therefore f ◦ T = f but f is non-constant. So T is not ergodic by theorem 1. Conversely let A be a measurable subset of S1 so that T −1(A) = A. Notice that the functions 1 1 n 2 en : S ! S defined by en(z) = z ; n 2 Z form an orthonormal basis for L (m). Let the P n Fourier series for χA be χA ∼ n bnen. Since en(T z) = a en(z), it follows by a change of variable that Z Z −n bn = χAe−n dm = a e−n dm; T −1(A) P n −1 and so χT −1(A) ∼ n a bnen. As T (A) = A, then χA = χT −1(A) and thus have the same n Fourier coefficients. Therefore bn = a bn for all n. If a is not a root of unity, the only way this can hold is if bn = 0 for all n 6= 0. By the uniqueness of Fourier coefficients, χA is a constant almost everywhere and so m(A) = 0 or 1. Therefore Ta is ergodic when a is not a root of unity. (3) Let ([0; 1); B; m) be the probability space consisting of the half open unit interval with Borel sets B and m the Lebesgue measure. Define T : [0; 1) ! [0; 1) by ( 2x; if 0 ≤ x < 1=2; T (x) = 2x mod 1 = 2x − 1; if 1=2 ≤ x < 1: This map is referred to as the dyadic transformation. Notice that if x has binary expansion x = 0:x1x2x3:::(2) then T (x) = 0:x2x3:::(2). It is worth showing that T is measure preserving. From measure theory [Billingsly, p 4], it suffices to prove that T preserves measure on a semi-algebra which generates the Borel σ-algebra. The collection of half open intervals with k j rational dyadic endpoints is such a semi-algebra. So let E = [ 2n ; 2n ) where n ≥ 0 and 3 Notes on the Ergodic Theorem 0 ≤ k ≤ j ≤ 2n.
Load more

Recommended publications
  • Deterministic Brownian Motion: the Effects of Perturbing a Dynamical
    1 Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System Michael C. Mackey∗and Marta Tyran-Kami´nska † October 26, 2018 Abstract Here we review and extend central limit theorems for highly chaotic but deterministic semi- dynamical discrete time systems. We then apply these results show how Brownian motion-like results are recovered, and how an Ornstein-Uhlenbeck process results within a totally determin- istic framework. These results illustrate that the contamination of experimental data by “noise” may, under certain circumstances, be alternately interpreted as the signature of an underlying chaotic process. Contents 1 Introduction 2 2 Semi-dynamical systems 5 2.1 Density evolution operators . ......... 5 2.2 Probabilistic and ergodic properties of density evolution ................ 12 2.3 Brownian motion from deterministic perturbations . ............... 16 2.3.1 Centrallimittheorems. ..... 16 2.3.2 FCLTfornoninvertiblemaps . ..... 20 2.4 Weak convergence criteria . ........ 31 3 Analysis 33 3.1 Weak convergence of v(tn) and vn. ............................ 35 3.2 Thelinearcaseinonedimension . ........ 36 3.2.1 Behaviour of the velocity variable . ......... 36 3.2.2 Behaviour of the position variable . ........ 38 4 Identifying the Limiting Velocity Distribution 41 4.1 Dyadicmap....................................... 42 arXiv:cond-mat/0408330v1 [cond-mat.stat-mech] 13 Aug 2004 4.2 Graphical illustration of the velocity density evolution with dyadic map perturbations 45 4.3 r-dyadicmap ..................................... 45 ∗e-mail: [email protected], Departments of Physiology, Physics & Mathematics and Centre for Nonlinear Dynamics, McGill University, 3655 Promenade Sir William Osler, Montreal, QC, CANADA, H3G 1Y6 †Corresponding author, email: [email protected], Institute of Mathematics, Silesian University, ul.
    [Show full text]
  • A Pseudo Random Numbers Generator Based on Chaotic Iterations
    A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking Christophe Guyeux, Qianxue Wang, Jacques Bahi To cite this version: Christophe Guyeux, Qianxue Wang, Jacques Bahi. A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking. WISM 2010, Int. Conf. on Web Information Systems and Mining, 2010, China. pp.202–211. hal-00563317 HAL Id: hal-00563317 https://hal.archives-ouvertes.fr/hal-00563317 Submitted on 4 Feb 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking Christophe Guyeux, Qianxue Wang, and Jacques M. Bahi University of Franche-Comte, Computer Science Laboratory LIFC, 25030 Besanc¸on Cedex, France {christophe.guyeux,qianxue.wang, jacques.bahi}@univ-fcomte.fr Abstract. In this paper, a new chaotic pseudo-random number generator (PRNG) is proposed. It combines the well-known ISAAC and XORshift generators with chaotic iterations. This PRNG possesses important properties of topological chaos and can successfully pass NIST and TestU01 batteries of tests. This makes our generator suitable for information security applications like cryptography. As an illustrative example, an application in the field of watermarking is presented.
    [Show full text]
  • Transformations)
    TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof.
    [Show full text]
  • On the Topological Conjugacy Problem for Interval Maps
    On the topological conjugacy problem for interval maps Roberto Venegeroles1, ∗ 1Centro de Matem´atica, Computa¸c~aoe Cogni¸c~ao,UFABC, 09210-170, Santo Andr´e,SP, Brazil (Dated: November 6, 2018) We propose an inverse approach for dealing with interval maps based on the manner whereby their branches are related (folding property), instead of addressing the map equations as a whole. As a main result, we provide a symmetry-breaking framework for determining topological conjugacy of interval maps, a well-known open problem in ergodic theory. Implications thereof for the spectrum and eigenfunctions of the Perron-Frobenius operator are also discussed. PACS numbers: 05.45.Ac, 02.30.Tb, 02.30.Zz Introduction - From the ergodic point of view, a mea- there exists a homeomorphism ! such that sure describing a chaotic map T is rightly one that, after S ! = ! T: (2) iterate a randomly chosen initial point, the iterates will ◦ ◦ be distributed according to this measure almost surely. Of course, the problem refers to the task of finding !, if This means that such a measure µ, called invariant, is it exists, when T and S are given beforehand. −1 preserved under application of T , i.e., µ[T (A)] = µ(A) Fundamental aspects of chaotic dynamics are pre- for any measurable subset A of the phase space [1, 2]. served under conjugacy. A map is chaotic if is topo- Physical measures are absolutely continuous with respect logical transitive and has sensitive dependence on initial to the Lebesgue measure, i.e., dµ(x) = ρ(x)dx, being the conditions. Both properties are preserved under conju- invariant density ρ(x) an eigenfunction of the Perron- gacy, then T is chaotic if and only if S is also.
    [Show full text]
  • Arxiv:2108.08461V1 [Math.DS] 19 Aug 2021 References 48
    THE BOOTSTRAP FOR DYNAMICAL SYSTEMS KASUN FERNANDO AND NAN ZOU Abstract. Despite their deterministic nature, dynamical systems often exhibit seemingly random behaviour. Consequently, a dynamical system is usually represented by a probabilistic model of which the unknown parameters must be estimated using statistical methods. When measuring the uncertainty of such parameter estimation, the bootstrap stands out as a simple but powerful technique. In this paper, we develop the bootstrap for dynamical systems and establish not only its consistency but also its second-order efficiency via a novel continuous Edgeworth expansion for dynamical systems. This is the first time such continuous Edgeworth expansions have been studied. Moreover, we verify the theoretical results about the bootstrap using computer simulations. Contents 1. Introduction 1 2. Notation and Preliminaries5 I Continuous Edgeworth Expansions 7 3. Spectral Assumptions8 4. First-order Continuous Edgeworth Expansions 19 5. Examples 24 II The Bootstrap 33 6. The Bootstrap Algorithm 33 7. Asymptotic Accuracy of the Bootstrap 35 8. Application of the Bootstrap to our Examples 39 9. Computer Simulation of the Bootstrap 42 arXiv:2108.08461v1 [math.DS] 19 Aug 2021 References 48 1. Introduction After its establishment in late 19th century through the efforts of Poincar´e[64] and Lyapunov [49], the theory of dynamical systems was applied to study the qualitative behaviour of dynamical processes in the real world. For example, the theory of dynamical systems has proved useful in, 2010 Mathematics Subject Classification. 62F40, 37A50, 62M05. Key words and phrases. dynamical systems, the bootstrap, continuous Edgeworth expansions, expanding maps, Markov chains. 1 2 KASUN FERNANDO AND NAN ZOU among many others, astronomy [12], chemical engineering [3], biology [51], ecology [74], demography [81], economics [75], language processing [65], neural activity [71], and machine learning [9], [16].
    [Show full text]
  • Contents Ilorpy
    From Deterministic Chaos to Anomalous Diffusion Dr. habil. Rainer Klages Queen Mary University of London School of Mathematical Sciences Mile End Road, London E1 4NS e-mail: [email protected] Contents 1 Introduction...................................... ........................ 2 2 Deterministic chaos................................ ....................... 3 2.1 Dynamicsofsimplemaps. .. .. 3 2.2 Ljapunovchaos .................................. 5 2.3 Entropies ..................................... 9 2.4 Opensystems, fractalsandescaperates. ....... 14 3 Deterministic diffusion............................. ....................... 19 3.1 What is deterministic diffusion? . ..... 20 3.2 Escape rate formalism for deterministic diffusion . ........... 22 3.2.1 Thediffusionequation . 22 arXiv:0804.3068v3 [nlin.CD] 20 Jul 2009 3.2.2 Basicideaoftheescaperateformalism . .... 24 3.2.3 The escape rate formalism worked out for a simple map . ...... 25 4 Anomalous diffusion................................. ..................... 29 4.1 Anomalous diffusion in intermittent maps . ...... 30 4.1.1 Whatisanomalousdiffusion? . 30 4.1.2 Continuoustimerandomwalktheory . 32 4.1.3 Afractionaldiffusionequation . 36 4.2 Anomalous diffusion of migrating biological cells . ........... 38 4.2.1 Cellmigration............................... 38 4.2.2 Experimentalresults . 39 4.2.3 Theoreticalmodeling . 41 5 Summary ........................................... ..................... 43 Bibliography....................................... ....................... 45 1 Introduction Over
    [Show full text]
  • One Or Two Small Points in Thermodynamics
    One or two small points in thermodynamics Walter F. Wreszinski∗ May 28, 2020 Abstract I present my recollections of what I used to find to be “one or two small points in thermodynamics”, following Sommerfeld’s famous quote, and review them on the light of present knowledge. 1 Introduction and motivation During my student days, I enjoyed the famous quote from Arnold Sommerfeld [Cal12]: “ Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, that it doesn’t bother you any more”. In fact, almost every student has found his “one or two small points”. This may be due to the fact that, in spite of being “the unique univer- sal science” [Had17], thermodynamics is populated with various logical and mathematical contradictions, to the point that - in another famous quote - the russian mathematician Vladimir I. Arnold was led to assert that “every mathematician knows it is impossible to understand an elementary course in thermodynamics” [Arn90]. arXiv:2003.03800v5 [cond-mat.stat-mech] 27 May 2020 Having written one such course (textbook) myself, I realized that I suc- ceeded only partially in trying to contradict Arnold’s quote, because, even after the completion of the second edition [Wre18], at least “two small points” did remain. The present note is my account of them.
    [Show full text]
  • Strongly Mixing Transformations and Geometric Diameters
    SARAJEVO JOURNAL OF MATHEMATICS DOI: 10.5644/SJM.08.2.06 Vol.8 (21) (2012), 245{257 STRONGLY MIXING TRANSFORMATIONS AND GEOMETRIC DIAMETERS HUSE FATKIC´ AND MEHMED BRKIC´ Dedicated to Professor Mustafa Kulenovi´con the occasion of his 60th birthday Abstract. In this paper, we investigate metric properties and disper- sive effects of strongly mixing transformations on general metric spaces endowed with a finite measure; in particular, we investigate their con- nections with the theory of generalized (geometric) diameters on general metric spaces. We first show that the known result by Rice [17,Theorem 2] (motivated by some physical phenomena and offer some clarifications of these phenomena), which is a substantial improvement of Theorems 1 and 2 due to Erber, Schweizer and Sklar [4], can be generalized in such a way that this result remains valid when \ordinary diameter" is replaced by \geometric diameter of any finite order". Next we show that \ordinary essential diameter" in the mentioned Rice's result can be replaced by \essential geometric diameter of any finite order". These results also complement the previous results of Fatki´c[ 6,8,10], Saff [18] and Sempi [20]. 1. Introduction and preliminaries In the broadest sense abstract dynamical systems and ergodic theory is the study of the qualitative properties of actions of groups on spaces (e.g. measure spaces, or topological spaces, or smooth manifolds). In this work we shall study actions of the group Z of integers on a measure space X, i.e., we study a transformation T : X ! X and its iterates T n, n 2 Z.
    [Show full text]
  • Some Examples in Ergodic Theory
    SOME EXAMPLES IN ERGODIC THEORY By YAEL NAIM DOWKER a& PBUL ERDBS [Received 22 February 1958.-Read 20 March 19581 Introduction THE purpose of this paper is to give some counter examples in ergodic theory. Some of the examples answer questions raised previously by some authors, some are counter examples to published theorems later withdrawn and some answer questions entertained privat,ely by the authors of this note. Let Y be an abstract space, y a a-field of subsets of Y and m a measure defined on y. We shall assume throughout that (Y, y, m) is a o-finite, non- atomic measure space. Let T be a 1: 1, ergodic measure preserving trans- formation of Y on to itself. Examples 1 and 2 deal with the case where (Y, y, a) is a finite measure space (in fact it is t,he ordinary line segment [O, 1). Example 1 shows that one cannot generalize theorems in Diophantine approximation t,heory like Theorem 2 (12, 76) to arbitrary ergodic conservative systems on [0, 1). Example 2 answers completely the questions raised by Halmos (7) and more recently by St’andish (13) concerning non-homogeneous ergodic theorems. We put and where fa(y) is the characteristic function of A. Examples 3, 4, and 5 and Theorems 1 and 2 deal with the case where m(Y) is infinite and study the relative behaviour of two ergodic measure preserving transformations T1 and T2. Example 3 shows that contrary to some expectations (1,2) the sequence -K(A, % $9 Y) = JfAG 4 Y)/~,(~, 4 Y) need not converge p.p.
    [Show full text]
  • NEW ADVANCES in SYMBOLIC DYNAMICS Bryna Kra. Symbolic
    NEW ADVANCES IN SYMBOLIC DYNAMICS MINI COURS Bryna Kra. Symbolic systems of low complexity This series of lectures will focus on shift systems of zero entropy, parti- cularly on the behaviors that arise when placing restrictions on the com- plexity and contrasting this behavior with that of general shifts. Starting with the simplest case of linear complexity, we discuss various properties of the system, such as the automorphism group and the number of inva- riant measures. We then discuss how these properties change as the growth rate becomes quadratic, a higher order polynomial, or higher growth, stu- dying both individual automorphisms and properties of the group of all automorphisms. Ronnie Pavlov. Entropy and mixing for multidimensional shifts of finite type I will speak about multidimensional shifts of finite type and their Gibbs measures. In particular, I will present results about computability of topo- logical entropy/pressure for SFTs and measure-theoretic entropy/pressure of Gibbs measures. I’ll focus on various mixing hypotheses, both topologi- cal and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square and Potts models. EXPOSÉS Shigeki Akiyama. On corona limits Given a tiling, we study the limit shape of successive coronas of the patches. It is shown that corona limit exists if and only if all directional speeds exist in all directions. If the directional speed are uniform in the sense of location, the corona limit is convex and symmetric with respect to the origin. Finally we see that if the tiling is periodic, all the conditions are satisfied and we see that the corona limit is convex symmetric polyhedron.
    [Show full text]
  • Dynamical Systems Notes
    Dynamical Systems Notes These are notes takento prepare for an oral exam in Dynamical Systems. The content comes from various sources, including from notes from a class taught by Arnd Scheel. Overview - Definitions : Dynamical System : Mathematical formalization for any fixed "rule" which describes the time dependence of a point’s position in its ambient space. The concept unifies very different types of such "rules" in mathematics: the different choices made for how time is measured and the special properties of the ambient space may give an idea of the vastness of the class of objects described by this concept. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the ambient space may be simply a set, without the need of a smooth space-time structure defined on it. There are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor. The measure theoretical definitions assume the existence of a measure-preserving transformation. This appears to exclude dissipative systems, as in a dissipative system a small region of phase space shrinks under time evolution. A simple construction (sometimes called the Krylov–Bogolyubov theorem) shows that it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.
    [Show full text]
  • Ergodic Theory and Linear Dynamics
    Introduction Gaussian measures Ergodic Gaussian measures for an operator How to find an ergodic measure Ergodic theory and linear dynamics Fr´ed´ericBayart Universit´eClermont-Ferrand 2 February 2011 Introduction Gaussian measures Ergodic Gaussian measures for an operator How to find an ergodic measure Ergodic theory Let (X ; B; µ) be a probability space and let T :(X ; B; µ) ! (X ; B; µ) be a measurable map. Definition (a) T is called a measure-preserving transformation if µ(T −1(A)) = µ(A) for all A 2 B. (b) T is called ergodic if it is measure-preserving and satisfies one of the following equivalent conditions: (i) Given any measurable sets A; B with positive measures, one can find an integer n ≥ 0 such that T n(A) \ B 6= ?; (ii) if A 2 B satisfies T (A) ⊂ A, then µ(A) = 0 or 1. Introduction Gaussian measures Ergodic Gaussian measures for an operator How to find an ergodic measure Example - Irrational rotations Let X = T, let B be the Borel σ-algebra on T and let µ be the Lebesgue measure on T. Let θ 2 RnQ. Then T :(X ; B; µ) ! (X ; B; µ) z 7! ei2πθz is ergodic. Introduction Gaussian measures Ergodic Gaussian measures for an operator How to find an ergodic measure Example - Dyadic transformation Let X = [0; 1), let B be the Borel σ-algebra on X and let µ be the Lebesgue measure on X . Then T :(X ; B; µ) ! (X ; B; µ) x 7! 2x mod 1 is ergodic. fx 2 X ;(xn;:::; xp) 2 En × · · · × Epg; Ej ⊂ {−1; 1g: µk is defined by µk (f1g) = 1=2 and µk ({−1g) = 1=2.
    [Show full text]