Measure, Topology, and Fractal Geometry

Die "Fraktale Geometrie" war das große Thema der Mathematik der 80er Jahre. Unter den zahlreichen Veröffentlichungen zu diesem Thema gibt es kaum ein Lehrbuch, das den Studenten gleichermaßen gründlich und anschaulich (16 Farbtafeln!) in dieses neue Gebiet einführt. "Es gibt keinen besseren Einstieg, um die Mathematik der fraktalen Geometrie zu lernen." Mathematics Teaching

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. The book provides a significant contribution to material already published about the beautiful mathematical objects drawn by computers using recursive or iterative algorithms. It nicely bridges the gap between traditional books on topology/analysis and more specialized treatises on fractal geometry. There are changes throughout the text in this updated edition, taking into account developments in the subject matter since 1990. Sections have been rewritten and are now clearer and more focused. The book also contains many good illustrations of fractals, including 16 color plates.

Bridges the gap between traditional books on topology/analysis and more specialized treatises on fractal geometry Contains plenty of examples, exercises, and illustrations Ideal for classroom use with a self-contained and careful presentation

Klappentext

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

Zusammenfassung

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.

Inhalt
Fractal Examples.- Metric Topology.- Topological Dimension.- Self-Similarity.- Measure Theory.- Fractal Dimension.- Additional Topics.