Measure, Topology, and Fractal Geometry (Undergraduate Texts in Mathematics)

From reviews of the first edition: ''In the world of mathematics, the 1980's might well be described as the ''decade of the fractal''. Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates).'' Mathematics Teaching ''The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples.'' Christoph Bandt, Mathematical Reviews ''...not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out.'' H.Haase, Zentralblatt About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused. For The Second Edition Of This Highly Regarded Textbook, Gerald Edgar Has Made Numerous Additions And Changes, In An Attempt To Provide A Clearer And More Focused Exposition. The Most Important Addition Is An Increased Emphasis On The Packing Measure, So That Now It Is Often Treated On A Par With The Hausdorff Measure. The Topological Dimensions Were Rearranged For Chapter 3, So That The Covering Dimension Is The Major One, And The Inductive Dimensions Are The Variants. A Reduced Cover Class Notion Was Introduced To Help In Proofs For Method I Or Method Ii Measures. Research Results Since 1990 That Affect These Elementary Topics Have Been Taken Into Account. Some Examples Have Been Added, Including Barnsley Leaf And Julia Set, And Most Of The Figures Have Been Re-drawn.^ From Reviews Of The First Edition: ...there Has Been A Deluge Of Books, Articles And Television Programmes About The Beautiful Mathematical Objects, Drawn By Computers Using Recursive Or Iterative Algorithms, Which Mandelbrot Christened Fractals. Gerald Edgar's Book Is A Significant Addition To This Deluge.^ Based On A Course Given To Talented High-school Students At Ohio University In 1988, It Is, In Fact, An Advanced Undergraduate Textbook About The Mathematics Of Fractal Geometry, Treating Such Topics As Metric Spaces, Measure Theory, Dimension Theory, And Even Some Algebraic Topology...the Book Also Contains Many Good Illustrations Of Fractals... - Mathematics Teaching The Book Can Be Recommended To Students Who Seriously Want To Know About The Mathematical Foundation Of Fractals, And To Lecturers Who Want To Illustrate A Standard Course In Metric Topology By Interesting Examples. - Christoph Bandt, Mathematical Reviews ...not Only Intended To Fit Mathematics Students Who Wish To Learn Fractal Geometry From Its Beginning But Also Students In Computer Science Who Are Interested In The Subject. [for Such Students] The Author Gives The Required Topics From Metric Topology And Measure Theory On An Elementary Level.^ The Book Is Written In A Very Clear Style And Contains A Lot Of Exercises Which Should Be Worked Out. - H.haase, Zentralblatt Fractal Examples -- Metric Topology -- Topological Dimension -- Self-similarity -- Measure Theory -- Fractal Dimension -- Additional Topics. Gerald Edgar. Includes Bibliographical References (p. 257-259) And Index. For the Second Edition of this highly regarded textbook, Gerald Edgar has made numerous additions and changes, in an attempt to provide a clearer and more focused exposition. The most important addition is an increased emphasis on the packing measure, so that now it is often treated on a par with the Hausdorff measure. The topological dimensions were rearranged for Chapter 3, so that the covering dimension is the major one, and the inductive dimensions are the variants. A "reduced cover class" notion was introduced to help in proofs for Method I or Method II measures. Research results since 1990 that affect these elementary topics have been taken into account. Some examples have been added, including Barnsley leaf and Julia set, and most of the figures have been re-drawn. From reviews of the First Edition: " ... there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high-school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology ... the book also contains many good illustrations of fractals ..."--Mathematics Teaching "The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples." - Christoph Bandt, Mathematical Reviews " ... not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. [For such students] the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out." - H. Haase, Zentralblatt Preface......Page 7 Contents......Page 14 The Triadic Cantor Dust......Page 17 The Sierpinski Gasket......Page 23 A Space of Strings......Page 27 Turtle Graphics......Page 30 Sets Defined Recursively......Page 34 Number Systems......Page 47 *Remarks......Page 51 Metric Space......Page 57 Metric Structures......Page 64 Separable and Compact Spaces......Page 73 Uniform Convergence......Page 81 The Hausdorff Metric......Page 87 Metrics for Strings......Page 91 *Remarks......Page 97 Zero-Dimensional Spaces......Page 101 Covering Dimension......Page 107 *Two-Dimensional Euclidean Space......Page 115 Inductive Dimension......Page 120 *Remarks......Page 129 Ratio Lists......Page 133 String Models......Page 138 Graph Self-Similarity......Page 141 *Remarks......Page 149 Lebesgue Measure......Page 153 Method I......Page 162 Two-Dimensional Lebesgue Measure......Page 168 Metric Outer Measure......Page 171 Measures for Strings......Page 175 *Remarks......Page 178 Hausdorff Measure......Page 181 Packing Measure......Page 185 Examples......Page 193 Self-Similarity......Page 201 The Open Set Condition......Page 206 Graph Self-Similarity......Page 215 Graph Open Set Condition......Page 221 *Other Fractal Dimensions......Page 226 *Remarks......Page 232 *Deconstruction......Page 241 *Self-Affine Sets......Page 245 *Self-Conformal......Page 250 *A Multifractal......Page 254 *A Superfractal......Page 258 *Remarks......Page 263 Terms......Page 267 Notation......Page 270 Reading......Page 271 References......Page 273 Index......Page 277 "For the Second Edition of this textbook, author Gerald Edgar has made numerous additions and changes, in an attempt to provide a clearer and more focused exposition. The most important addition is an increased emphasis on the packing measure, so that now it is often treated on a par with the Hausdorff measure. The topological dimensions were rearranged for Chapter 3, so that the covering dimension is the major one, and the inductive dimensions are the variants. A "reduced cover class" notion was introduced to help in proofs for Method I or Method II measures. Research results since 1990 that affect these elementary topics have been taken into account. Some examples have been added, including Barnsley leaf and Julia set, and most of the figures have been re-drawn."--Jacket Based on a course given to talented high-school students at Ohio University in 1988, this book is essentially an advanced undergraduate textbook about the mathematics of fractal geometry. It nicely bridges the gap between traditional books on topology/analysis and more specialized treatises on fractal geometry. The book treats such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. It takes into account developments in the subject matter since 1990. Sections are clear and focused. The book contains plenty of examples, exercises, and good illustrations of fractals, including 16 color plates. This book provides the mathematics necessary for the study of fractal geometry. It includes background material on metric topology and measure theory and also covers topological and fractal dimension, including the Hausdorff dimension. Furthermore, the book contains a complete discussion of self-similarity as well as the more general "graph self-similarity."