Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C
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Effective S-Adic Symbolic Dynamical Systems
Effective S-adic symbolic dynamical systems Val´erieBerth´e,Thomas Fernique, and Mathieu Sablik? 1 IRIF, CNRS UMR 8243, Univ. Paris Diderot, France [email protected] 2 LIPN, CNRS UMR 7030, Univ. Paris 13, France [email protected] 3 I2M UMR 7373, Aix Marseille Univ., France [email protected] Abstract. We focus in this survey on effectiveness issues for S-adic sub- shifts and tilings. An S-adic subshift or tiling space is a dynamical system obtained by iterating an infinite composition of substitutions, where a substitution is a rule that replaces a letter by a word (that might be multi-dimensional), or a tile by a finite union of tiles. Several notions of effectiveness exist concerning S-adic subshifts and tiling spaces, such as the computability of the sequence of iterated substitutions, or the effec- tiveness of the language. We compare these notions and discuss effective- ness issues concerning classical properties of the associated subshifts and tiling spaces, such as the computability of shift-invariant measures and the existence of local rules (soficity). We also focus on planar tilings. Keywords: Symbolic dynamics; adic map; substitution; S-adic system; planar tiling; local rules; sofic subshift; subshift of finite type; computable invariant measure; effective language. 1 Introduction Decidability in symbolic dynamics and ergodic theory has already a long history. Let us quote as an illustration the undecidability of the emptiness problem (the domino problem) for multi-dimensional subshifts of finite type (SFT) [8, 40], or else the connections between effective ergodic theory, computable analysis and effective randomness (see for instance [14, 33, 44]). -
Ergodic Theory
MATH41112/61112 Ergodic Theory Charles Walkden 4th January, 2018 MATH4/61112 Contents Contents 0 Preliminaries 2 1 An introduction to ergodic theory. Uniform distribution of real se- quences 4 2 More on uniform distribution mod 1. Measure spaces. 13 3 Lebesgue integration. Invariant measures 23 4 More examples of invariant measures 38 5 Ergodic measures: definition, criteria, and basic examples 43 6 Ergodic measures: Using the Hahn-Kolmogorov Extension Theorem to prove ergodicity 53 7 Continuous transformations on compact metric spaces 62 8 Ergodic measures for continuous transformations 72 9 Recurrence 83 10 Birkhoff’s Ergodic Theorem 89 11 Applications of Birkhoff’s Ergodic Theorem 99 12 Solutions to the Exercises 108 1 MATH4/61112 0. Preliminaries 0. Preliminaries 0.1 Contact details § The lecturer is Dr Charles Walkden, Room 2.241, Tel: 0161 275 5805, Email: [email protected]. My office hour is: Monday 2pm-3pm. If you want to see me at another time then please email me first to arrange a mutually convenient time. 0.2 Course structure § This is a reading course, supported by one lecture per week. I have split the notes into weekly sections. You are expected to have read through the material before the lecture, and then go over it again afterwards in your own time. In the lectures I will highlight the most important parts, explain the statements of the theorems and what they mean in practice, and point out common misunderstandings. As a general rule, I will not normally go through the proofs in great detail (but they are examinable unless indicated otherwise). -
Strictly Ergodic Symbolic Dynamical Systems
STRICTLY ERGODIC SYMBOLIC DYNAMICAL SYSTEMS SHIZUO KAKUTANI YALE UNIVERSITY 1. Introduction We continue the study of strictly ergodic symbolic dynamical systems which was started in our earlier report [6]. The main tools used in this investigation are "homomorphisms" and "substitutions". Among other things, we construct two strictly ergodic symbolic dynamical systems which are weakly mixing but not strongly mixing. 2. Strictly ergodic symbolic dynamical systems Let A be a finite set consisting of more than one element. Let (2.1) X = AZ = H A, An = A forallne Z, neZ be the set of all two sided infinite sequences (2.2) x = {a"In Z}, an = A for all n E Z, where (2.3) Z = {nln = O, + 1, + 2,} is the set of all integers. For each n E Z, an is called the nth coordinate of x, and the mapping (2.4) 7r,: x -+ a, = 7En(x) is called the nth projection of the power space X = AZ onto the base space An = A. The space X is a totally disconnected, compact, metrizable space with respect to the usual direct product topology. Let q be a one to one mapping of X = AZ onto itself defined by (2.5) 7En(q(X)) = ir.+1(X) for all n E Z. The mapping p is a homeomorphism ofX onto itself and is called the shift trans- formation. The dynamical system (X, (p) thus obtained is called the shift dynamical system. This research was supported in part by NSF Grant GP16392. 319 320 SIXTH BERKELEY SYMPOSIUM: KAKUTANI Let X0 be a nonempty closed subset of X which is invariant under (p. -
MA427 Ergodic Theory
MA427 Ergodic Theory Course Notes (2012-13) 1 Introduction 1.1 Orbits Let X be a mathematical space. For example, X could be the unit interval [0; 1], a circle, a torus, or something far more complicated like a Cantor set. Let T : X ! X be a function that maps X into itself. Let x 2 X be a point. We can repeatedly apply the map T to the point x to obtain the sequence: fx; T (x);T (T (x));T (T (T (x))); : : : ; :::g: We will often write T n(x) = T (··· (T (T (x)))) (n times). The sequence of points x; T (x);T 2(x);::: is called the orbit of x. We think of applying the map T as the passage of time. Thus we think of T (x) as where the point x has moved to after time 1, T 2(x) is where the point x has moved to after time 2, etc. Some points x 2 X return to where they started. That is, T n(x) = x for some n > 1. We say that such a point x is periodic with period n. By way of contrast, points may move move densely around the space X. (A sequence is said to be dense if (loosely speaking) it comes arbitrarily close to every point of X.) If we take two points x; y of X that start very close then their orbits will initially be close. However, it often happens that in the long term their orbits move apart and indeed become dramatically different. -
Conformal Fractals – Ergodic Theory Methods
Conformal Fractals – Ergodic Theory Methods Feliks Przytycki Mariusz Urba´nski May 17, 2009 2 Contents Introduction 7 0 Basic examples and definitions 15 1 Measure preserving endomorphisms 25 1.1 Measurespacesandmartingaletheorem . 25 1.2 Measure preserving endomorphisms, ergodicity . .. 28 1.3 Entropyofpartition ........................ 34 1.4 Entropyofendomorphism . 37 1.5 Shannon-Mcmillan-Breimantheorem . 41 1.6 Lebesguespaces............................ 44 1.7 Rohlinnaturalextension . 48 1.8 Generalized entropy, convergence theorems . .. 54 1.9 Countabletoonemaps.. .. .. .. .. .. .. .. .. .. 58 1.10 Mixingproperties . .. .. .. .. .. .. .. .. .. .. 61 1.11 Probabilitylawsand Bernoulliproperty . 63 Exercises .................................. 68 Bibliographicalnotes. 73 2 Compact metric spaces 75 2.1 Invariantmeasures . .. .. .. .. .. .. .. .. .. .. 75 2.2 Topological pressure and topological entropy . ... 83 2.3 Pressureoncompactmetricspaces . 87 2.4 VariationalPrinciple . 89 2.5 Equilibrium states and expansive maps . 94 2.6 Functionalanalysisapproach . 97 Exercises ..................................106 Bibliographicalnotes. 109 3 Distance expanding maps 111 3.1 Distance expanding open maps, basic properties . 112 3.2 Shadowingofpseudoorbits . 114 3.3 Spectral decomposition. Mixing properties . 116 3.4 H¨older continuous functions . 122 3 4 CONTENTS 3.5 Markov partitions and symbolic representation . 127 3.6 Expansive maps are expanding in some metric . 134 Exercises .................................. 136 Bibliographicalnotes. 138 4 -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Turbulence, Entropy and Dynamics
TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions -
AN INTRODUCTION to DYNAMICAL BILLIARDS Contents 1
AN INTRODUCTION TO DYNAMICAL BILLIARDS SUN WOO PARK 2 Abstract. Some billiard tables in R contain crucial references to dynamical systems but can be analyzed with Euclidean geometry. In this expository paper, we will analyze billiard trajectories in circles, circular rings, and ellipses as well as relate their charactersitics to ergodic theory and dynamical systems. Contents 1. Background 1 1.1. Recurrence 1 1.2. Invariance and Ergodicity 2 1.3. Rotation 3 2. Dynamical Billiards 4 2.1. Circle 5 2.2. Circular Ring 7 2.3. Ellipse 9 2.4. Completely Integrable 14 Acknowledgments 15 References 15 Dynamical billiards exhibits crucial characteristics related to dynamical systems. Some billiard tables in R2 can be understood with Euclidean geometry. In this ex- pository paper, we will analyze some of the billiard tables in R2, specifically circles, circular rings, and ellipses. In the first section we will present some preliminary background. In the second section we will analyze billiard trajectories in the afore- mentioned billiard tables and relate their characteristics with dynamical systems. We will also briefly discuss the notion of completely integrable billiard mappings and Birkhoff's conjecture. 1. Background (This section follows Chapter 1 and 2 of Chernov [1] and Chapter 3 and 4 of Rudin [2]) In this section, we define basic concepts in measure theory and ergodic theory. We will focus on probability measures, related theorems, and recurrent sets on certain maps. The definitions of probability measures and σ-algebra are in Chapter 1 of Chernov [1]. 1.1. Recurrence. Definition 1.1. Let (X,A,µ) and (Y ,B,υ) be measure spaces. -
Symbolic Dynamics and Markov Partitions
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 1, January 1998, Pages 1–56 S 0273-0979(98)00737-X SYMBOLIC DYNAMICS AND MARKOV PARTITIONS ROY L. ADLER Abstract. The decimal expansion of real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by sym- bolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by sym- bolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely, hyperbolic automorphisms of the two dimensional torus. While there are some results in higher dimensions, this area remains a fertile one for research. 1. Introduction We address the question: how and to what extent can a dynamical system be represented by a symbolic one? The first use of infinite sequences of symbols to describe orbits is attributed to a nineteenth century work of Hadamard [H]. How- ever the present work is rooted in something very much older and familiar to us all: namely, the representation of real numbers by infinite binary expansions. As the example of Section 3.2 shows, such a representation is related to the behavior of a special partition under the action of a map. Partitions of this sort have been linked to the name Markov because of their connection to discrete time Markov processes. -
Ergodic Theory, Fractal Tops and Colour Stealing 1
ERGODIC THEORY, FRACTAL TOPS AND COLOUR STEALING MICHAEL BARNSLEY Abstract. A new structure that may be associated with IFS and superIFS is described. In computer graphics applications this structure can be rendered using a new algorithm, called the “colour stealing”. 1. Ergodic Theory and Fractal Tops The goal of this lecture is to describe informally some recent realizations and work in progress concerning IFS theory with application to the geometric modelling and assignment of colours to IFS fractals and superfractals. The results will be described in the simplest setting of a single IFS with probabilities, but many generalizations are possible, most notably to superfractals. Let the iterated function system (IFS) be denoted (1.1) X; f1, ..., fN ; p1,...,pN . { } This consists of a finite set of contraction mappings (1.2) fn : X X,n=1, 2, ..., N → acting on the compact metric space (1.3) (X,d) with metric d so that for some (1.4) 0 l<1 ≤ we have (1.5) d(fn(x),fn(y)) l d(x, y) ≤ · for all x, y X.Thepn’s are strictly positive probabilities with ∈ (1.6) pn =1. n X The probability pn is associated with the map fn. We begin by reviewing the two standard structures (one and two)thatare associated with the IFS 1.1, namely its set attractor and its measure attractor, with emphasis on the Collage Property, described below. This property is of particular interest for geometrical modelling and computer graphics applications because it is the key to designing IFSs with attractor structures that model given inputs. -
Dynamical Systems and Ergodic Theory
MATH36206 - MATHM6206 Dynamical Systems and Ergodic Theory Teaching Block 1, 2017/18 Lecturer: Prof. Alexander Gorodnik PART III: LECTURES 16{30 course web site: people.maths.bris.ac.uk/∼mazag/ds17/ Copyright c University of Bristol 2010 & 2016. This material is copyright of the University. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only. Chapter 3 Ergodic Theory In this last part of our course we will introduce the main ideas and concepts in ergodic theory. Ergodic theory is a branch of dynamical systems which has strict connections with analysis and probability theory. The discrete dynamical systems f : X X studied in topological dynamics were continuous maps f on metric spaces X (or more in general, topological→ spaces). In ergodic theory, f : X X will be a measure-preserving map on a measure space X (we will see the corresponding definitions below).→ While the focus in topological dynamics was to understand the qualitative behavior (for example, periodicity or density) of all orbits, in ergodic theory we will not study all orbits, but only typical1 orbits, but will investigate more quantitative dynamical properties, as frequencies of visits, equidistribution and mixing. An example of a basic question studied in ergodic theory is the following. Let A X be a subset of O+ ⊂ the space X. Consider the visits of an orbit f (x) to the set A. If we consider a finite orbit segment x, f(x),...,f n−1(x) , the number of visits to A up to time n is given by { } Card 0 k n 1, f k(x) A . -
Ergodic Theory Approach to Chaos: Remarks and Computational Aspects
Int. J. Appl. Math. Comput. Sci., 2012, Vol. 22, No. 2, 259–267 DOI: 10.2478/v10006-012-0019-4 ERGODIC THEORY APPROACH TO CHAOS: REMARKS AND COMPUTATIONAL ASPECTS ∗ ∗∗ PAWEŁ J. MITKOWSKI ,WOJCIECH MITKOWSKI ∗ Faculty of Electrical Engineering, Automatics, Computer Science and Electronics AGH University of Science and Technology, al. Mickiewicza 30/B-1, 30-059 Cracow, Poland e-mail: [email protected] ∗∗Department of Automatics AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland e-mail: [email protected] We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the pre- viously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota’s conjecture concerning nontrivial ergodic properties of the model. Keywords: ergodic theory, chaos, invariant measures, attractors, delay differential equations. 1. Introduction man, 2001). Transformations (or flows) with an invariant measure display three main levels of irregular behaviour, In the literature concerning dynamical systems we can i.e., (ranging from the lowest to the highest) ergodicity, find many definitions of chaos in various approaches mixing and exactness. Between ergodicity and mixing we (Rudnicki, 2004; Devaney, 1987; Bronsztejn et al., 2004). can also distinguish light mixing, mild mixing and weak Our central issue here will be the ergodic theory appro- mixing (Lasota and Mackey, 1994; Silva, 2010) and, on ach.