Three hours THE UNIVERSITY OF MANCHESTER. 31st May :00 17:00
|
|
- Brandon Hunter
- 5 years ago
- Views:
Transcription
1 Three hours MATH41112 THE UNIVERSITY OF MANCHESTER ERGODIC THEORY 31st May :00 17:00 Answer FOUR of the FIVE questions. If more than four questions are attempted, then credit will be given for the best four answers. Electronic calculators are permitted, provided they cannot store text. 1 of 6 P.T.O.
2 1. (i) Let x n R be a sequence of real numbers. What does it mean to say that x n is uniformly distributed mod 1? (ii) Recall that Weyl s criterion says that the following two statements are equivalent: (a) the sequence x n R is uniformly distributed mod 1; (b) for each l Z \ {0}, we have 1 n 1 lim e 2πilx j = 0. n n j=0 Prove that (b) (a). (You may use without proof the fact that trigonometric polynomials are uniformly dense in the space of real-valued continuous functions defined on R/Z.) [14 marks] (iii) Let α, β R. Define the sequence x n by x n = αn + β. Suppose that α Q. Use Weyl s Criterion to prove that x n is uniformly distributed mod 1. (iv) Let x n be a sequence. Let m 1 and define the sequence x (m) n of mth differences by x (m) n = x n+m x n. It was proved in the course that if, for every m 1, the sequence of mth differences is uniformly distributed mod 1 then x n is uniformly distributed mod 1. Using this result and (iii) above, prove that x n = αn 2 + βn + γ is uniformly distributed mod 1 if α Q. (v) Are either of the sequences n 2 + 2n + 1 n 2 + n + 2 [6 marks] uniformly distributed mod 1? Justify your answers. 2 of 6 P.T.O.
3 2. (i) Let T be a measurable transformation of the probability space (X, B, µ). What does it mean to say that µ is an invariant measure for T? (ii) Define T : [0, 1] [0, 1] by T (x) = 4x(1 x). Show that T does not preserve Lebesgue measure. Define a probability measure µ on [0, 1] by µ(b) = 1 π B dx x(1 x). Let [a, b] [0, 1]. Show that ([ 1 1 a µ, 1 ]) 1 b = 1 µ([a, b]) and hence show that µ is a T -invariant measure. [16 marks] (iii) Let X R 3 denote the sphere with centre (0, 0, 1) and radius 1, let S = (0, 0, 0) denote the south pole and let N = (0, 0, 2) denote the north pole. Define the stereographic projection map φ : X \ {N} R 2 {0} in the following way: for x X \ {N} draw the straight line through N and x and define φ(x) to be the unique point of intersection between this line and R 2 {0} R 3, as illustrated in Figure 1. Figure 1: See Question 2(iii). Define T : X X by T (x) = φ 1 ( 1 2 φ(x)) for x N and T (N) = N. Then T is a homeomorphism of X (you do not need to prove this). Determine all invariant Borel probability measures for T. [12 marks] 3 of 6 P.T.O.
4 3. (i) Let (X, B, µ) be a probability space and let T : X X be a measurable transformation. Suppose that µ is a T -invariant measure. What does it mean to say that T is ergodic with respect to µ? (ii) Let α, β R. Define T : R 2 /Z 2 R 2 /Z 2 : (x, y) + Z 2 (x + α, x + y + β) + Z 2. Show that T is ergodic with respect to Lebesgue measure µ if and only if α Q. (You may use any standard characterisations of ergodicity from the course, provided that you state them clearly. You may also assume without proof that µ is a T -invariant measure.) [16 marks] (iii) Let S = {1,..., k} and let Σ = {x = (x j ) j=0 x j S}. Let σ : Σ Σ denote the shift map: σ(x 0, x 1, x 2,...) = (x 1, x 2, x 3,...). For i j S, define cylinder sets by [i 0, i 1,..., i n 1 ] = {x = (x j ) j=0 x j = i j, 0 j n 1}. Let (p(1),..., p(k)) be a probability vector (so that p(j) > 0 and k j=1 p(j) = 1). Define a Bernoulli measure µ by defining it on cylinders by µ([i 0, i 1,..., i n 1 ]) = p(i 1 )p(i 2 ) p(i n 1 ) and then extending it to the Borel σ-algebra using the Hahn-Kolmogorov Extension Theorem. One can prove that µ is a σ-invariant measure (you do not need to do this). Let I, J be two cylinders. Prove that provided that n is sufficiently large. µ(i σ n J) = µ(i)µ(j) Hence prove that σ is ergodic with respect to µ. (You may use without proof the fact that if there exists K > 0 such that µ(b)µ(i) Kµ(B I) for all cylinders I then µ(b) = 0 or 1.) [12 marks] 4 of 6 P.T.O.
5 4. Let X be a compact metric space, let B denote the Borel σ-algebra, and let T : X X be continuous. Let M(X) denote the space of Borel probability measures on X. Let C(X, R) denote the space of all continuous real-valued functions defined on X. (i) What does it mean to say that a sequence of measures µ n M(X) weak* converges to µ M(X)? (ii) The measure T µ is defined, for µ M(X), by (T µ)(b) = µ(t 1 B). (You may assume without proof that T µ M(X).) Prove that integration with respect to T µ is given by the formula f d(t µ) = f T dµ for f L 1 (X, B, µ). [6 marks] (iii) Prove that the map T : M(X) M(X) is weak* continuous. Prove that the space M(X, T ) of all T -invariant Borel probability measures is a weak* closed subset of M(X). [6 marks] (iv) State, without proof, the Riesz Representation Theorem. Prove that the following two conditions are equivalent: (i) µ M(X) is T -invariant; (ii) for all f C(X, R) we have f T dµ = f dµ. [8 marks] (v) Suppose that T : [0, 1] [0, 1] is continuous. Prove that there exists some x [0, 1] such that δ x is T -invariant. Does this remain true if [0, 1] is replaced by the circle R/Z? [8 marks] 5 of 6 P.T.O.
6 5. (i) Let (X, B, µ) be a probability space and let T : X X be an ergodic measure-preserving transformation. Let f L 1 (X, B, µ). State, without proof, Birkhoff s Ergodic Theorem. (ii) Suppose that T is ergodic with respect to µ and let A B. Use Birkhoff s Ergodic Theorem to show that the frequency with which the orbit of µ-almost every point x X visits A is equal to µ(a). (iii) Let (X, B, µ) be a probability space and let T : X X be an ergodic measure-preserving transformation. Let f L 1 (X, B, µ). Let A, B B. Deduce from Birkhoff s Ergodic Theorem and the definition of the Lebesgue integral that (a) implies (b) in the following: (a) T is ergodic with respect to µ (b) for all f, g L 2 (X, B, µ) we have 1 n 1 lim n n j=0 f(t j x)g(x) dµ = f dµ g dµ. (1) (One can also prove that (b) implies (a) but you are not required to do this.) [12 marks] (iv) Suppose that for all f, g L 2 (X, B, µ) we have f(t n x)g(x) dµ = lim n Prove that T is ergodic with respect to µ. f dµ g dµ. (2) Let α R be irrational. Recall that the irrational circle rotation T : R/Z R/Z, T (x) = x + α mod 1 is ergodic with respect to Lebesgue measure (you do not need to prove this). Give an example of a pair of L 2 functions f, g to show, however, that (2) does not hold for T. (Hint: consider exponential functions.) [8 marks] (v) Suppose that (X, B, µ) is a probability space and T : X X is a measure-preserving, but not necessarily ergodic, transformation. Let A, B B be arbitrary sets and suppose that µ(a), µ(b) > 0. Is it necessarily true that for µ-almost every point x of A there are infinitely many n for which T n (x) A? Is it necessarily true that for µ-almost every point x of B there are infinitely many n for which T n (x) A? In each case justify your answer by either quoting a theorem from the course or by giving a counter-example. END OF EXAMINATION PAPER 6 of 6
3 hours UNIVERSITY OF MANCHESTER. 22nd May and. Electronic calculators may be used, provided that they cannot store text.
3 hours MATH40512 UNIVERSITY OF MANCHESTER DYNAMICAL SYSTEMS AND ERGODIC THEORY 22nd May 2007 9.45 12.45 Answer ALL four questions in SECTION A (40 marks in total) and THREE of the four questions in SECTION
More information13. Examples of measure-preserving tranformations: rotations of a torus, the doubling map
3. Examples of measure-preserving tranformations: rotations of a torus, the doubling map 3. Rotations of a torus, the doubling map In this lecture we give two methods by which one can show that a given
More information4. Ergodicity and mixing
4. Ergodicity and mixing 4. Introduction In the previous lecture we defined what is meant by an invariant measure. In this lecture, we define what is meant by an ergodic measure. The primary motivation
More informationThree hours THE UNIVERSITY OF MANCHESTER. 24th January
Three hours MATH41011 THE UNIVERSITY OF MANCHESTER FOURIER ANALYSIS AND LEBESGUE INTEGRATION 24th January 2013 9.45 12.45 Answer ALL SIX questions in Section A (25 marks in total). Answer THREE of the
More informationMATH41112/ Ergodic Theory. Charles Walkden
MATH42/62 Ergodic Theory Charles Walkden 4 th January, 208 MATH4/62 Contents Contents 0 Preliminaries 2 An introduction to ergodic theory. Uniform distribution of real sequences 4 2 More on uniform distribution
More informationRudiments of Ergodic Theory
Rudiments of Ergodic Theory Zefeng Chen September 24, 203 Abstract In this note we intend to present basic ergodic theory. We begin with the notion of a measure preserving transformation. We then define
More informationDYNAMICAL SYSTEMS PROBLEMS. asgor/ (1) Which of the following maps are topologically transitive (minimal,
DYNAMICAL SYSTEMS PROBLEMS http://www.math.uci.edu/ asgor/ (1) Which of the following maps are topologically transitive (minimal, topologically mixing)? identity map on a circle; irrational rotation of
More informationUNIVERSITY OF BRISTOL. Mock exam paper for examination for the Degrees of B.Sc. and M.Sci. (Level 3)
UNIVERSITY OF BRISTOL Mock exam paper for examination for the Degrees of B.Sc. and M.Sci. (Level 3) DYNAMICAL SYSTEMS and ERGODIC THEORY MATH 36206 (Paper Code MATH-36206) 2 hours and 30 minutes This paper
More informationProbability and Measure
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability
More informationLecture Notes Introduction to Ergodic Theory
Lecture Notes Introduction to Ergodic Theory Tiago Pereira Department of Mathematics Imperial College London Our course consists of five introductory lectures on probabilistic aspects of dynamical systems,
More information10. The ergodic theory of hyperbolic dynamical systems
10. The ergodic theory of hyperbolic dynamical systems 10.1 Introduction In Lecture 8 we studied thermodynamic formalism for shifts of finite type by defining a suitable transfer operator acting on a certain
More informationErgodic Theory. Constantine Caramanis. May 6, 1999
Ergodic Theory Constantine Caramanis ay 6, 1999 1 Introduction Ergodic theory involves the study of transformations on measure spaces. Interchanging the words measurable function and probability density
More information25.1 Ergodicity and Metric Transitivity
Chapter 25 Ergodicity This lecture explains what it means for a process to be ergodic or metrically transitive, gives a few characterizes of these properties (especially for AMS processes), and deduces
More informationAverage laws in analysis
Average laws in analysis Silvius Klein Norwegian University of Science and Technology (NTNU) The law of large numbers: informal statement The theoretical expected value of an experiment is approximated
More informationIntegration on Measure Spaces
Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of
More informationVARIATIONAL PRINCIPLE FOR THE ENTROPY
VARIATIONAL PRINCIPLE FOR THE ENTROPY LUCIAN RADU. Metric entropy Let (X, B, µ a measure space and I a countable family of indices. Definition. We say that ξ = {C i : i I} B is a measurable partition if:
More informationDynamical Systems and Ergodic Theory PhD Exam Spring Topics: Topological Dynamics Definitions and basic results about the following for maps and
Dynamical Systems and Ergodic Theory PhD Exam Spring 2012 Introduction: This is the first year for the Dynamical Systems and Ergodic Theory PhD exam. As such the material on the exam will conform fairly
More informationDisintegration into conditional measures: Rokhlin s theorem
Disintegration into conditional measures: Rokhlin s theorem Let Z be a compact metric space, µ be a Borel probability measure on Z, and P be a partition of Z into measurable subsets. Let π : Z P be the
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More informationTHE STRUCTURE OF THE SPACE OF INVARIANT MEASURES
THE STRUCTURE OF THE SPACE OF INVARIANT MEASURES VAUGHN CLIMENHAGA Broadly, a dynamical system is a set X with a map f : X is discrete time. Continuous time considers a flow ϕ t : Xö. mostly consider discrete
More informationProblem Set 6: Solutions Math 201A: Fall a n x n,
Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series
More informationarxiv: v2 [math.ds] 24 Apr 2018
CONSTRUCTION OF SOME CHOWLA SEQUENCES RUXI SHI arxiv:1804.03851v2 [math.ds] 24 Apr 2018 Abstract. For numerical sequences taking values 0 or complex numbers of modulus 1, we define Chowla property and
More informationMath 240 (Driver) Qual Exam (9/12/2017)
1 Name: I.D. #: Math 240 (Driver) Qual Exam (9/12/2017) Instructions: Clearly explain and justify your answers. You may cite theorems from the text, notes, or class as long as they are not what the problem
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationErgodic Theory and Topological Groups
Ergodic Theory and Topological Groups Christopher White November 15, 2012 Throughout this talk (G, B, µ) will denote a measure space. We call the space a probability space if µ(g) = 1. We will also assume
More informationNORMAL NUMBERS AND UNIFORM DISTRIBUTION (WEEKS 1-3) OPEN PROBLEMS IN NUMBER THEORY SPRING 2018, TEL AVIV UNIVERSITY
ORMAL UMBERS AD UIFORM DISTRIBUTIO WEEKS -3 OPE PROBLEMS I UMBER THEORY SPRIG 28, TEL AVIV UIVERSITY Contents.. ormal numbers.2. Un-natural examples 2.3. ormality and uniform distribution 2.4. Weyl s criterion
More informationMATH 104 : Final Exam
MATH 104 : Final Exam 10 May, 2017 Name: You have 3 hours to answer the questions. You are allowed one page (front and back) worth of notes. The page should not be larger than a standard US letter size.
More informationApplication of Ergodic Theory to Uniform distribution mod 1. Oleg Ivrii
Application of Ergodic Theory to Uniform distribution mod 1 Oleg Ivrii February 13, 2008 Chapter 1 Ergodic Theory 1.1 The Setting Our setting will vary, but (, µ) will be some measure space and T a measure
More informationIRRATIONAL ROTATION OF THE CIRCLE AND THE BINARY ODOMETER ARE FINITARILY ORBIT EQUIVALENT
IRRATIONAL ROTATION OF THE CIRCLE AND THE BINARY ODOMETER ARE FINITARILY ORBIT EQUIVALENT MRINAL KANTI ROYCHOWDHURY Abstract. Two invertible dynamical systems (X, A, µ, T ) and (Y, B, ν, S) where X, Y
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationMATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION. Extra Reading Material for Level 4 and Level 6
MATH41011/MATH61011: FOURIER SERIES AND LEBESGUE INTEGRATION Extra Reading Material for Level 4 and Level 6 Part A: Construction of Lebesgue Measure The first part the extra material consists of the construction
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationTHEOREMS, ETC., FOR MATH 516
THEOREMS, ETC., FOR MATH 516 Results labeled Theorem Ea.b.c (or Proposition Ea.b.c, etc.) refer to Theorem c from section a.b of Evans book (Partial Differential Equations). Proposition 1 (=Proposition
More informationProof: The coding of T (x) is the left shift of the coding of x. φ(t x) n = L if T n+1 (x) L
Lecture 24: Defn: Topological conjugacy: Given Z + d (resp, Zd ), actions T, S a topological conjugacy from T to S is a homeomorphism φ : M N s.t. φ T = S φ i.e., φ T n = S n φ for all n Z + d (resp, Zd
More information9 Radon-Nikodym theorem and conditioning
Tel Aviv University, 2015 Functions of real variables 93 9 Radon-Nikodym theorem and conditioning 9a Borel-Kolmogorov paradox............. 93 9b Radon-Nikodym theorem.............. 94 9c Conditioning.....................
More informationA VERY BRIEF REVIEW OF MEASURE THEORY
A VERY BRIEF REVIEW OF MEASURE THEORY A brief philosophical discussion. Measure theory, as much as any branch of mathematics, is an area where it is important to be acquainted with the basic notions and
More informationMATH & MATH FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING Scientia Imperii Decus et Tutamen 1
MATH 5310.001 & MATH 5320.001 FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING 2016 Scientia Imperii Decus et Tutamen 1 Robert R. Kallman University of North Texas Department of Mathematics 1155
More informationconsists of two disjoint copies of X n, each scaled down by 1,
Homework 4 Solutions, Real Analysis I, Fall, 200. (4) Let be a topological space and M be a σ-algebra on which contains all Borel sets. Let m, µ be two positive measures on M. Assume there is a constant
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationMAGIC010 Ergodic Theory Lecture Entropy
7. Entropy 7. Introduction A natural question in mathematics is the so-called isomorphism problem : when are two mathematical objects of the same class the same (in some appropriately defined sense of
More informationLebesgue Measure. Dung Le 1
Lebesgue Measure Dung Le 1 1 Introduction How do we measure the size of a set in IR? Let s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its
More informationMeasures and Measure Spaces
Chapter 2 Measures and Measure Spaces In summarizing the flaws of the Riemann integral we can focus on two main points: 1) Many nice functions are not Riemann integrable. 2) The Riemann integral does not
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationMath 461 Homework 8. Paul Hacking. November 27, 2018
Math 461 Homework 8 Paul Hacking November 27, 2018 (1) Let S 2 = {(x, y, z) x 2 + y 2 + z 2 = 1} R 3 be the sphere with center the origin and radius 1. Let N = (0, 0, 1) S 2 be the north pole. Let F :
More informationMath 461 Homework 8 Paul Hacking November 27, 2018
(1) Let Math 461 Homework 8 Paul Hacking November 27, 2018 S 2 = {(x, y, z) x 2 +y 2 +z 2 = 1} R 3 be the sphere with center the origin and radius 1. Let N = (0, 0, 1) S 2 be the north pole. Let F : S
More informationPart II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationDynamical Systems and Ergodic Theory
Dynamical Systems and 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
More informationMEASURABLE DYNAMICS OF SIMPLE p-adic POLYNOMIALS JOHN BRYK AND CESAR E. SILVA
MEASURABLE DYNAMICS OF SIMPLE p-adic POLYNOMIALS JOHN BRYK AND CESAR E. SILVA 1. INTRODUCTION. The p-adic numbers have many fascinating properties that are different from those of the real numbers. These
More informationLyapunov optimizing measures for C 1 expanding maps of the circle
Lyapunov optimizing measures for C 1 expanding maps of the circle Oliver Jenkinson and Ian D. Morris Abstract. For a generic C 1 expanding map of the circle, the Lyapunov maximizing measure is unique,
More informationMATH 5616H INTRODUCTION TO ANALYSIS II SAMPLE FINAL EXAM: SOLUTIONS
MATH 5616H INTRODUCTION TO ANALYSIS II SAMPLE FINAL EXAM: SOLUTIONS You may not use notes, books, etc. Only the exam paper, a pencil or pen may be kept on your desk during the test. Calculators are not
More information7. Let X be a (general, abstract) metric space which is sequentially compact. Prove X must be complete.
Math 411 problems The following are some practice problems for Math 411. Many are meant to challenge rather that be solved right away. Some could be discussed in class, and some are similar to hard exam
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationComplex Analysis MATH 6300 Fall 2013 Homework 4
Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,
More informationAnnalee Gomm Math 714: Assignment #2
Annalee Gomm Math 714: Assignment #2 3.32. Verify that if A M, λ(a = 0, and B A, then B M and λ(b = 0. Suppose that A M with λ(a = 0, and let B be any subset of A. By the nonnegativity and monotonicity
More informationMATH 614 Dynamical Systems and Chaos Lecture 38: Ergodicity (continued). Mixing.
MATH 614 Dynamical Systems and Chaos Lecture 38: Ergodicity (continued). Mixing. Ergodic theorems Let (X,B,µ) be a measured space and T : X X be a measure-preserving transformation. Birkhoff s Ergodic
More informationMeasurable functions are approximately nice, even if look terrible.
Tel Aviv University, 2015 Functions of real variables 74 7 Approximation 7a A terrible integrable function........... 74 7b Approximation of sets................ 76 7c Approximation of functions............
More information3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first
Math 632/6321: Theory of Functions of a Real Variable Sample Preinary Exam Questions 1. Let (, M, µ) be a measure space. (a) Prove that if µ() < and if 1 p < q
More informationSOLUTIONS TO SOME PROBLEMS
23 FUNCTIONAL ANALYSIS Spring 23 SOLUTIONS TO SOME PROBLEMS Warning:These solutions may contain errors!! PREPARED BY SULEYMAN ULUSOY PROBLEM 1. Prove that a necessary and sufficient condition that the
More informationIn terms of measures: Exercise 1. Existence of a Gaussian process: Theorem 2. Remark 3.
1. GAUSSIAN PROCESSES A Gaussian process on a set T is a collection of random variables X =(X t ) t T on a common probability space such that for any n 1 and any t 1,...,t n T, the vector (X(t 1 ),...,X(t
More informationfunctions as above. There is a unique non-empty compact set, i=1
1 Iterated function systems Many of the well known examples of fractals, including the middle third Cantor sets, the Siepiński gasket and certain Julia sets, can be defined as attractors of iterated function
More informationIndependent random variables
CHAPTER 2 Independent random variables 2.1. Product measures Definition 2.1. Let µ i be measures on (Ω i,f i ), 1 i n. Let F F 1... F n be the sigma algebra of subsets of Ω : Ω 1... Ω n generated by all
More informationTHE RADON NIKODYM PROPERTY AND LIPSCHITZ EMBEDDINGS
THE RADON NIKODYM PROPERTY AND LIPSCHITZ EMBEDDINGS Motivation The idea here is simple. Suppose we have a Lipschitz homeomorphism f : X Y where X and Y are Banach spaces, namely c 1 x y f (x) f (y) c 2
More information2 Probability, random elements, random sets
Tel Aviv University, 2012 Measurability and continuity 25 2 Probability, random elements, random sets 2a Probability space, measure algebra........ 25 2b Standard models................... 30 2c Random
More informationMath General Topology Fall 2012 Homework 8 Solutions
Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that
More informationApplied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.
Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationTHE INVERSE FUNCTION THEOREM
THE INVERSE FUNCTION THEOREM W. PATRICK HOOPER The implicit function theorem is the following result: Theorem 1. Let f be a C 1 function from a neighborhood of a point a R n into R n. Suppose A = Df(a)
More informationPreliminary Exam 2018 Solutions to Morning Exam
Preliminary Exam 28 Solutions to Morning Exam Part I. Solve four of the following five problems. Problem. Consider the series n 2 (n log n) and n 2 (n(log n)2 ). Show that one converges and one diverges
More information1 Orthonormal sets in Hilbert space
Math 857 Fall 15 1 Orthonormal sets in Hilbert space Let S H. We denote by [S] the span of S, i.e., the set of all linear combinations of elements from S. A set {u α : α A} is called orthonormal, if u
More informationAnalysis Qualifying Exam
Analysis Qualifying Exam Spring 2017 Problem 1: Let f be differentiable on R. Suppose that there exists M > 0 such that f(k) M for each integer k, and f (x) M for all x R. Show that f is bounded, i.e.,
More informationFinal. due May 8, 2012
Final due May 8, 2012 Write your solutions clearly in complete sentences. All notation used must be properly introduced. Your arguments besides being correct should be also complete. Pay close attention
More informationS-adic sequences A bridge between dynamics, arithmetic, and geometry
S-adic sequences A bridge between dynamics, arithmetic, and geometry J. M. Thuswaldner (joint work with P. Arnoux, V. Berthé, M. Minervino, and W. Steiner) Marseille, November 2017 REVIEW OF PART 1 Sturmian
More informationFACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355. Analysis 4. Examiner: Professor S. W. Drury Date: Wednesday, April 18, 2007 INSTRUCTIONS
FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355 Analysis 4 Examiner: Professor S. W. Drury Date: Wednesday, April 18, 27 Associate Examiner: Professor K. N. GowriSankaran Time: 2: pm. 5: pm.
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationMath 240 (Driver) Qual Exam (5/22/2017)
1 Name: I.D. #: Math 240 (Driver) Qual Exam (5/22/2017) Instructions: Clearly explain and justify your answers. You may cite theorems from the text, notes, or class as long as they are not what the problem
More information1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0
4 INUTES. If, ω, ω, -----, ω 9 are the th roots of unity, then ( + ω) ( + ω ) ----- ( + ω 9 ) is B) D) 5. i If - i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for
More informationIntroduction to Functional Analysis
Introduction to Functional Analysis Carnegie Mellon University, 21-640, Spring 2014 Acknowledgements These notes are based on the lecture course given by Irene Fonseca but may differ from the exact lecture
More information2 hours THE UNIVERSITY OF MANCHESTER.?? January 2017??:????:??
hours MATH3051 THE UNIVERSITY OF MANCHESTER HYPERBOLIC GEOMETRY?? January 017??:????:?? Answer ALL FOUR questions in Section A (40 marks in all) and TWO of the THREE questions in Section B (30 marks each).
More informationFolland: Real Analysis, Chapter 7 Sébastien Picard
Folland: Real Analysis, Chapter 7 Sébastien Picard Problem 7.2 Let µ be a Radon measure on X. a. Let N be the union of all open U X such that µ(u) =. Then N is open and µ(n) =. The complement of N is called
More informationI. ANALYSIS; PROBABILITY
ma414l1.tex Lecture 1. 12.1.2012 I. NLYSIS; PROBBILITY 1. Lebesgue Measure and Integral We recall Lebesgue measure (M411 Probability and Measure) λ: defined on intervals (a, b] by λ((a, b]) := b a (so
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationERGODIC DYNAMICAL SYSTEMS CONJUGATE TO THEIR COMPOSITION SQUARES. 0. Introduction
Acta Math. Univ. Comenianae Vol. LXXI, 2(2002), pp. 201 210 201 ERGODIC DYNAMICAL SYSTEMS CONJUGATE TO THEIR COMPOSITION SQUARES G. R. GOODSON Abstract. We investigate the question of when an ergodic automorphism
More information(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M.
1. Abstract Integration The main reference for this section is Rudin s Real and Complex Analysis. The purpose of developing an abstract theory of integration is to emphasize the difference between the
More information= F (b) F (a) F (x i ) F (x i+1 ). a x 0 x 1 x n b i
Real Analysis Problem 1. If F : R R is a monotone function, show that F T V ([a,b]) = F (b) F (a) for any interval [a, b], and that F has bounded variation on R if and only if it is bounded. Here F T V
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More informationA Short Introduction to Ergodic Theory of Numbers. Karma Dajani
A Short Introduction to Ergodic Theory of Numbers Karma Dajani June 3, 203 2 Contents Motivation and Examples 5 What is Ergodic Theory? 5 2 Number Theoretic Examples 6 2 Measure Preserving, Ergodicity
More informationCOBHAM S THEOREM AND SUBSTITUTION SUBSHIFTS
COBHAM S THEOREM AND SUBSTITUTION SUBSHIFTS FABIEN DURAND Abstract. This lecture intends to propose a first contact with subshift dynamical systems through the study of a well known family: the substitution
More informationKatznelson Problems. Prakash Balachandran Duke University. June 19, 2009
Katznelson Problems Prakash Balachandran Duke University June 9, 9 Chapter. Compute the Fourier coefficients of the following functions (defined by their values on [ π, π)): f(t) { t < t π (t) g(t) { t
More informationMath 190: Fall 2014 Homework 4 Solutions Due 5:00pm on Friday 11/7/2014
Math 90: Fall 04 Homework 4 Solutions Due 5:00pm on Friday /7/04 Problem : Recall that S n denotes the n-dimensional unit sphere: S n = {(x 0, x,..., x n ) R n+ : x 0 + x + + x n = }. Let N S n denote
More informationLectures on dynamical systems and entropy
Lectures on dynamical systems and entropy Michael Hochman June 27, 204 Please report any errors to mhochman@math.huji.ac.il Contents Introduction 4 2 Measure preserving transformations 6 2. Measure preserving
More informationNotes for the course: Ergodic theory and entropy.
Notes for the course: Ergodic theory and entropy. Barney Bramham Contents Bochum Summer Semester 204 Preliminaries from measure theory and functional analysis 5. Measures..............................
More informationOn fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems
On fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems Jose Cánovas, Jiří Kupka* *) Institute for Research and Applications of Fuzzy Modeling University of Ostrava Ostrava, Czech
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationCHAPTER V DUAL SPACES
CHAPTER V DUAL SPACES DEFINITION Let (X, T ) be a (real) locally convex topological vector space. By the dual space X, or (X, T ), of X we mean the set of all continuous linear functionals on X. By the
More informationQuantitative recurrence for beta expansion. Wang BaoWei
Introduction Further study Quantitative recurrence for beta expansion Huazhong University of Science and Technology July 8 2010 Introduction Further study Contents 1 Introduction Background Beta expansion
More informationSeminar In Topological Dynamics
Bar Ilan University Department of Mathematics Seminar In Topological Dynamics EXAMPLES AND BASIC PROPERTIES Harari Ronen May, 2005 1 2 1. Basic functions 1.1. Examples. To set the stage, we begin with
More information