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Lectures on Fractal Geometry and Dynamics
Lectures on fractal geometry and dynamics Michael Hochman∗ March 25, 2012 Contents 1 Introduction 2 2 Preliminaries 3 3 Dimension 3 3.1 A family of examples: Middle-α Cantor sets . 3 3.2 Minkowski dimension . 4 3.3 Hausdorff dimension . 9 4 Using measures to compute dimension 13 4.1 The mass distribution principle . 13 4.2 Billingsley's lemma . 15 4.3 Frostman's lemma . 18 4.4 Product sets . 22 5 Iterated function systems 24 5.1 The Hausdorff metric . 24 5.2 Iterated function systems . 26 5.3 Self-similar sets . 31 5.4 Self-affine sets . 36 6 Geometry of measures 39 6.1 The Besicovitch covering theorem . 39 6.2 Density and differentiation theorems . 45 6.3 Dimension of a measure at a point . 48 6.4 Upper and lower dimension of measures . 50 ∗Send comments to [email protected] 1 6.5 Hausdorff measures and their densities . 52 1 Introduction Fractal geometry and its sibling, geometric measure theory, are branches of analysis which study the structure of \irregular" sets and measures in metric spaces, primarily d R . The distinction between regular and irregular sets is not a precise one but informally, k regular sets might be understood as smooth sub-manifolds of R , or perhaps Lipschitz graphs, or countable unions of the above; whereas irregular sets include just about 1 everything else, from the middle- 3 Cantor set (still highly structured) to arbitrary d Cantor sets (irregular, but topologically the same) to truly arbitrary subsets of R . d For concreteness, let us compare smooth sub-manifolds and Cantor subsets of R . -
Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
Version of 21.8.15 Chapter 43 Topologies and Measures II The
Version of 21.8.15 Chapter 43 Topologies and measures II The first chapter of this volume was ‘general’ theory of topological measure spaces; I attempted to distinguish the most important properties a topological measure can have – inner regularity, τ-additivity – and describe their interactions at an abstract level. I now turn to rather more specialized investigations, looking for features which offer explanations of the behaviour of the most important spaces, radiating outwards from Lebesgue measure. In effect, this chapter consists of three distinguishable parts and two appendices. The first three sections are based on ideas from descriptive set theory, in particular Souslin’s operation (§431); the properties of this operation are the foundation for the theory of two classes of topological space of particular importance in measure theory, the K-analytic spaces (§432) and the analytic spaces (§433). The second part of the chapter, §§434-435, collects miscellaneous results on Borel and Baire measures, looking at the ways in which topological properties of a space determine properties of the measures it carries. In §436 I present the most important theorems on the representation of linear functionals by integrals; if you like, this is the inverse operation to the construction of integrals from measures in §122. The ideas continue into §437, where I discuss spaces of signed measures representing the duals of spaces of continuous functions, and topologies on spaces of measures. The first appendix, §438, looks at a special topic: the way in which the patterns in §§434-435 are affected if we assume that our spaces are not unreasonably complex in a rather special sense defined in terms of measures on discrete spaces. -
Measure and Integration (Under Construction)
Measure and Integration (under construction) Gustav Holzegel∗ April 24, 2017 Abstract These notes accompany the lecture course ”Measure and Integration” at Imperial College London (Autumn 2016). They follow very closely the text “Real-Analysis” by Stein-Shakarchi, in fact most proofs are simple rephrasings of the proofs presented in the aforementioned book. Contents 1 Motivation 3 1.1 Quick review of the Riemann integral . 3 1.2 Drawbacks of the class R, motivation of the Lebesgue theory . 4 1.2.1 Limits of functions . 5 1.2.2 Lengthofcurves ....................... 5 1.2.3 TheFundamentalTheoremofCalculus . 5 1.3 Measures of sets in R ......................... 6 1.4 LiteratureandFurtherReading . 6 2 Measure Theory: Lebesgue Measure in Rd 7 2.1 Preliminaries and Notation . 7 2.2 VolumeofRectanglesandCubes . 7 2.3 Theexteriormeasure......................... 9 2.3.1 Examples ........................... 9 2.3.2 Propertiesoftheexteriormeasure . 10 2.4 Theclassof(Lebesgue)measurablesets . 12 2.4.1 The property of countable additivity . 14 2.4.2 Regularity properties of the Lebesgue measure . 15 2.4.3 Invariance properties of the Lebesgue measure . 16 2.4.4 σ-algebrasandBorelsets . 16 2.5 Constructionofanon-measurableset. 17 2.6 MeasurableFunctions ........................ 19 2.6.1 Some abstract preliminary remarks . 19 2.6.2 Definitions and equivalent formulations of measurability . 19 2.6.3 Properties 1: Behaviour under compositions . 21 2.6.4 Properties 2: Behaviour under limits . 21 2.6.5 Properties3: Behaviourof sums and products . 22 ∗Imperial College London, Department of Mathematics, South Kensington Campus, Lon- don SW7 2AZ, United Kingdom. 1 2.6.6 Thenotionof“almosteverywhere”. 22 2.7 Building blocks of integration theory . -
Pontrjagin Duality for Finite Groups
PONTRJAGIN DUALITY FOR FINITE GROUPS Partha Sarathi Chakraborty The Institute of Mathematical Sciences Chennai e1; ··· ; en : basis for V . ∗ V := fφjφ : V ! C linear map g. ∗ hφi ; ej i := φi (ej ) := δij , φi ; ··· ; φn, dual basis for V . Dual of a map, T : V ! W T ∗ : W ∗ ! V ∗; T ∗(φ)(v) = φ(T (v)). T : V ! W ; S : W ! U; (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ ! W ∗ Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The finite group case Pontryagin Duality Motivation: duality for vector spaces Dual of a vector space V : a finite dimensional vector space over C. ∗ V := fφjφ : V ! C linear map g. ∗ hφi ; ej i := φi (ej ) := δij , φi ; ··· ; φn, dual basis for V . Dual of a map, T : V ! W T ∗ : W ∗ ! V ∗; T ∗(φ)(v) = φ(T (v)). T : V ! W ; S : W ! U; (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ ! W ∗ Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The finite group case Pontryagin Duality Motivation: duality for vector spaces Dual of a vector space V : a finite dimensional vector space over C. e1; ··· ; en : basis for V . ∗ hφi ; ej i := φi (ej ) := δij , φi ; ··· ; φn, dual basis for V . Dual of a map, T : V ! W T ∗ : W ∗ ! V ∗; T ∗(φ)(v) = φ(T (v)). T : V ! W ; S : W ! U; (S ◦ T )∗ = T ∗ ◦ S∗ : U∗ ! W ∗ Pontryagin duality: classical formulation Motivation from linear algebra Limitations of the classical formulation dual of a cyclic group Some algebraic structures duals for direct sums and products The finite group case Pontryagin Duality Motivation: duality for vector spaces Dual of a vector space V : a finite dimensional vector space over C. -
Extremal Dependence Concepts
Extremal dependence concepts Giovanni Puccetti1 and Ruodu Wang2 1Department of Economics, Management and Quantitative Methods, University of Milano, 20122 Milano, Italy 2Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada Journal version published in Statistical Science, 2015, Vol. 30, No. 4, 485{517 Minor corrections made in May and June 2020 Abstract The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. -
1 the Basic Set-Up 2 Poisson Brackets
MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016–2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical Mechanics, Chapter 14. 1 The basic set-up I assume that you have already studied Gregory, Sections 14.1–14.4. The following is intended only as a succinct summary. We are considering a system whose equations of motion are written in Hamiltonian form. This means that: 1. The phase space of the system is parametrized by canonical coordinates q =(q1,...,qn) and p =(p1,...,pn). 2. We are given a Hamiltonian function H(q, p, t). 3. The dynamics of the system is given by Hamilton’s equations of motion ∂H q˙i = (1a) ∂pi ∂H p˙i = − (1b) ∂qi for i =1,...,n. In these notes we will consider some deeper aspects of Hamiltonian dynamics. 2 Poisson brackets Let us start by considering an arbitrary function f(q, p, t). Then its time evolution is given by n df ∂f ∂f ∂f = q˙ + p˙ + (2a) dt ∂q i ∂p i ∂t i=1 i i X n ∂f ∂H ∂f ∂H ∂f = − + (2b) ∂q ∂p ∂p ∂q ∂t i=1 i i i i X 1 where the first equality used the definition of total time derivative together with the chain rule, and the second equality used Hamilton’s equations of motion. The formula (2b) suggests that we make a more general definition. Let f(q, p, t) and g(q, p, t) be any two functions; we then define their Poisson bracket {f,g} to be n def ∂f ∂g ∂f ∂g {f,g} = − . -
Arithmetic Duality Theorems
Arithmetic Duality Theorems Second Edition J.S. Milne Copyright c 2004, 2006 J.S. Milne. The electronic version of this work is licensed under a Creative Commons Li- cense: http://creativecommons.org/licenses/by-nc-nd/2.5/ Briefly, you are free to copy the electronic version of the work for noncommercial purposes under certain conditions (see the link for a precise statement). Single paper copies for noncommercial personal use may be made without ex- plicit permission from the copyright holder. All other rights reserved. First edition published by Academic Press 1986. A paperback version of this work is available from booksellers worldwide and from the publisher: BookSurge, LLC, www.booksurge.com, 1-866-308-6235, [email protected] BibTeX information @book{milne2006, author={J.S. Milne}, title={Arithmetic Duality Theorems}, year={2006}, publisher={BookSurge, LLC}, edition={Second}, pages={viii+339}, isbn={1-4196-4274-X} } QA247 .M554 Contents Contents iii I Galois Cohomology 1 0 Preliminaries............................ 2 1 Duality relative to a class formation . ............. 17 2 Localfields............................. 26 3 Abelianvarietiesoverlocalfields.................. 40 4 Globalfields............................. 48 5 Global Euler-Poincar´echaracteristics................ 66 6 Abelianvarietiesoverglobalfields................. 72 7 An application to the conjecture of Birch and Swinnerton-Dyer . 93 8 Abelianclassfieldtheory......................101 9 Otherapplications..........................116 AppendixA:Classfieldtheoryforfunctionfields............126 -
MARSTRAND's THEOREM and TANGENT MEASURES Contents 1
MARSTRAND'S THEOREM AND TANGENT MEASURES ALEKSANDER SKENDERI Abstract. This paper aims to prove and motivate Marstrand's theorem, which is a fundamental result in geometric measure theory. Along the way, we introduce and motivate the notion of tangent measures, while also proving several related results. The paper assumes familiarity with the rudiments of measure theory such as Lebesgue measure, Lebesgue integration, and the basic convergence theorems. Contents 1. Introduction 1 2. Preliminaries on Measure Theory 2 2.1. Weak Convergence of Measures 2 2.2. Differentiation of Measures and Covering Theorems 3 2.3. Hausdorff Measure and Densities 4 3. Marstrand's Theorem 5 3.1. Tangent measures and uniform measures 5 3.2. Proving Marstrand's Theorem 8 Acknowledgments 16 References 16 1. Introduction Some of the most important objects in all of mathematics are smooth manifolds. Indeed, they occupy a central position in differential geometry, algebraic topology, and are often very important when relating mathematics to physics. However, there are also many objects of interest which may not be differentiable over certain sets of points in their domains. Therefore, we want to study more general objects than smooth manifolds; these objects are known as rectifiable sets. The following definitions and theorems require the notion of Hausdorff measure. If the reader is unfamiliar with Hausdorff measure, he should consult definition 2.10 of section 2.3 before continuing to read this section. Definition 1.1. A k-dimensional Borel set E ⊂ Rn is called rectifiable if there 1 k exists a countable family fΓigi=1 of Lipschitz graphs such that H (E n [ Γi) = 0. -
Poisson Structures and Integrability
Poisson Structures and Integrability Peter J. Olver University of Minnesota http://www.math.umn.edu/ olver ∼ Hamiltonian Systems M — phase space; dim M = 2n Local coordinates: z = (p, q) = (p1, . , pn, q1, . , qn) Canonical Hamiltonian system: dz O I = J H J = − dt ∇ ! I O " Equivalently: dpi ∂H dqi ∂H = = dt − ∂qi dt ∂pi Lagrange Bracket (1808): n ∂pi ∂qi ∂qi ∂pi [ u , v ] = ∂u ∂v − ∂u ∂v i#= 1 (Canonical) Poisson Bracket (1809): n ∂u ∂v ∂u ∂v u , v = { } ∂pi ∂qi − ∂qi ∂pi i#= 1 Given functions u , . , u , the (2n) (2n) matrices with 1 2n × respective entries [ u , u ] u , u i, j = 1, . , 2n i j { i j } are mutually inverse. Canonical Poisson Bracket n ∂F ∂H ∂F ∂H F, H = F T J H = { } ∇ ∇ ∂pi ∂qi − ∂qi ∂pi i#= 1 = Poisson (1809) ⇒ Hamiltonian flow: dz = z, H = J H dt { } ∇ = Hamilton (1834) ⇒ First integral: dF F, H = 0 = 0 F (z(t)) = const. { } ⇐⇒ dt ⇐⇒ Poisson Brackets , : C∞(M, R) C∞(M, R) C∞(M, R) { · · } × −→ Bilinear: a F + b G, H = a F, H + b G, H { } { } { } F, a G + b H = a F, G + b F, H { } { } { } Skew Symmetric: F, H = H, F { } − { } Jacobi Identity: F, G, H + H, F, G + G, H, F = 0 { { } } { { } } { { } } Derivation: F, G H = F, G H + G F, H { } { } { } F, G, H C∞(M, R), a, b R. ∈ ∈ In coordinates z = (z1, . , zm), F, H = F T J(z) H { } ∇ ∇ where J(z)T = J(z) is a skew symmetric matrix. − The Jacobi identity imposes a system of quadratically nonlinear partial differential equations on its entries: ∂J jk ∂J ki ∂J ij J il + J jl + J kl = 0 ! ∂zl ∂zl ∂zl " #l Given a Poisson structure, the Hamiltonian flow corresponding to H C∞(M, R) is the system of ordinary differential equati∈ons dz = z, H = J(z) H dt { } ∇ Lie’s Theory of Function Groups Used for integration of partial differential equations: F , F = G (F , . -
Measure Theory John K. Hunter
Measure Theory John K. Hunter Department of Mathematics, University of California at Davis Abstract. These are some brief notes on measure theory, concentrating on n Lebesgue measure on R . Some missing topics I would have liked to have in- cluded had time permitted are: the change of variable formula for the Lebesgue n integral on R ; absolutely continuous functions and functions of bounded vari- ation of a single variable and their connection with Lebesgue-Stieltjes measures n on R; Radon measures on R , and other locally compact Hausdorff topological spaces, and the Riesz representation theorem for bounded linear functionals on spaces of continuous functions; and other examples of measures, including n k-dimensional Hausdorff measure in R , Wiener measure and Brownian mo- tion, and Haar measure on topological groups. All these topics can be found in the references. c John K. Hunter, 2011 Contents Chapter 1. Measures 1 1.1. Sets 1 1.2. Topological spaces 2 1.3. Extended real numbers 2 1.4. Outer measures 3 1.5. σ-algebras 4 1.6. Measures 5 1.7. Sets of measure zero 6 Chapter 2. Lebesgue Measure on Rn 9 2.1. Lebesgue outer measure 10 2.2. Outer measure of rectangles 12 2.3. Carath´eodory measurability 14 2.4. Null sets and completeness 18 2.5. Translational invariance 19 2.6. Borel sets 20 2.7. Borel regularity 22 2.8. Linear transformations 27 2.9. Lebesgue-Stieltjes measures 30 Chapter 3. Measurable functions 33 3.1. Measurability 33 3.2. Real-valued functions 34 3.3. -
Dirichlet Is Natural Vincent Danos, Ilias Garnier
Dirichlet is Natural Vincent Danos, Ilias Garnier To cite this version: Vincent Danos, Ilias Garnier. Dirichlet is Natural. MFPS 31 - Mathematical Foundations of Program- ming Semantics XXXI, Jun 2015, Nijmegen, Netherlands. pp.137-164, 10.1016/j.entcs.2015.12.010. hal-01256903 HAL Id: hal-01256903 https://hal.archives-ouvertes.fr/hal-01256903 Submitted on 15 Jan 2016 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. MFPS 2015 Dirichlet is natural Vincent Danos1 D´epartement d'Informatique Ecole normale sup´erieure Paris, France Ilias Garnier2 School of Informatics University of Edinburgh Edinburgh, United Kingdom Abstract Giry and Lawvere's categorical treatment of probabilities, based on the probabilistic monad G, offer an elegant and hitherto unexploited treatment of higher-order probabilities. The goal of this paper is to follow this formulation to reconstruct a family of higher-order probabilities known as the Dirichlet process. This family is widely used in non-parametric Bayesian learning. Given a Polish space X, we build a family of higher-order probabilities in G(G(X)) indexed by M ∗(X) the set of non-zero finite measures over X. The construction relies on two ingredients.