UID:
almafu_9960117740702883
Format:
1 online resource (ix, 402 pages) :
,
digital, PDF file(s).
ISBN:
1-316-86692-0
,
1-316-86818-4
,
1-316-46023-1
Series Statement:
Cambridge studies in advanced mathematics ; 162
Content:
This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic.
Note:
Title from publisher's bibliographic system (viewed on 31 Jan 2017).
,
Cover -- Half-title -- Series page -- Frontispiece -- Title page -- Copyright information -- Table of contents -- Preface -- 1 Minkowski and Hausdorff dimensions -- 1.1 Minkowski dimension -- 1.2 Hausdorff dimension and the Mass Distribution Principle -- 1.3 Sets defined by digit restrictions -- 1.4 Billingsley's Lemma and the dimension of measures -- 1.5 Sets defined by digit frequency -- 1.6 Slices -- 1.7 Intersecting translates of Cantor sets -- 1.8 Notes -- 1.9 Exercises -- 2 Self-similarity and packing dimension -- 2.1 Self-similar sets -- 2.2 The open set condition is sufficient -- 2.3 Homogeneous sets -- 2.4 Microsets -- 2.5 Poincare sets -- 2.6 Alternative definitions of Minkowski dimension -- 2.7 Packing measures and dimension -- 2.8 When do packing and Minkowski dimension agree? -- 2.9 Notes -- 2.10 Exercises -- 3 Frostman's theory and capacity -- 3.1 Frostman's Lemma -- 3.2 The dimension of product sets -- 3.3 Generalized Marstrand Slicing Theorem -- 3.4 Capacity and dimension -- 3.5 Marstrand's Projection Theorem -- 3.6 Mapping a tree to Euclidean space preserves capacity -- 3.7 Dimension of random Cantor sets -- 3.8 Notes -- 3.9 Exercises -- 4 Self-affine sets -- 4.1 Construction and Minkowski dimension -- 4.2 The Hausdorff dimension of self-affine sets -- 4.3 A dichotomy for Hausdorff measure -- 4.4 The Hausdorff measure is infinite -- 4.5 Notes -- 4.6 Exercises -- 5 Graphs of continuous functions -- 5.1 Holder continuous functions -- 5.2 The Weierstrass function is nowhere differentiable -- 5.3 Lower Holder estimates -- 5.4 Notes -- 5.5 Exercises -- 6 Brownian motion, Part I -- 6.1 Gaussian random variables -- 6.2 Levy's construction of Brownian motion -- 6.3 Basic properties of Brownian motion -- 6.4 Hausdorff dimension of the Brownian path and graph -- 6.5 Nowhere differentiability is prevalent.
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6.6 Strong Markov property and the reflection principle -- 6.7 Local extrema of Brownian motion -- 6.8 Area of planar Brownian motion -- 6.9 General Markov processes -- 6.10 Zeros of Brownian motion -- 6.11 Harris' inequality and its consequences -- 6.12 Points of increase -- 6.13 Notes -- 6.14 Exercises -- 7 Brownian motion, Part II -- 7.1 Dimension doubling -- 7.2 The Law of the Iterated Logarithm -- 7.3 Skorokhod's Representation -- 7.3.1 Root's Method * -- 7.3.2 Dubins' Stopping Rule * -- 7.3.3 Skorokhod's representation for a sequence -- 7.4 Donsker's Invariance Principle -- 7.5 Harmonic functions and Brownian motion in R< -- sub> -- d< -- sub> -- -- 7.6 The maximum principle for harmonic functions -- 7.7 The Dirichlet problem -- 7.8 Polar points and recurrence -- 7.9 Conformal invariance * -- 7.10 Capacity and harmonic functions -- 7.11 Notes -- 7.12 Exercises -- 8 Random walks, Markov chains and capacity -- 8.1 Frostman's theory for discrete sets -- 8.2 Markov chains and capacity -- 8.3 Intersection equivalence and return times -- 8.4 Lyons' Theorem on percolation on trees -- 8.5 Dimension of random Cantor sets (again) -- 8.6 Brownian motion and Martin capacity -- 8.7 Notes -- 8.8 Exercises -- 9 Besicovitch-Kakeya sets -- 9.1 Existence and dimension -- 9.2 Splitting triangles -- 9.3 Fefferman's Disk Multiplier Theorem * -- 9.4 Random Besicovitch sets -- 9.5 Projections of self-similar Cantor sets -- 9.6 The open set condition is necessary * -- 9.7 Notes -- 9.8 Exercises -- 10 The Traveling Salesman Theorem -- 10.1 Lines and length -- 10.2 The β -numbers -- 10.3 Counting with dyadic squares -- 10.4 β and μ are equivalent -- 10.5 β -sums estimate minimal paths -- 10.6 Notes -- 10.7 Exercises -- Appendix A Banach's Fixed-Point Theorem -- A.1 Banach's Fixed-Point Theorem -- A.2 Blaschke Selection Theorem -- A.3 Dual Lipschitz metric.
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Appendix B Frostman's Lemma for analytic sets -- B.1 Borel sets are analytic -- B.2 Choquet capacitability -- Appendix C Hints and solutions to selected exercises -- References -- Index.
Additional Edition:
ISBN 1-107-13411-0
Language:
English
URL:
https://doi.org/10.1017/9781316460238
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