# Fractal-Based Methods in Analysis

Springer Science & Business Media, Nov 18, 2011 - Mathematics - 408 pages

The idea of modeling the behaviour of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems.

"Fractal-Based Methods in Analysis" draws together, for the first time in book form, methods and results from almost twenty years of research in this topic, including new viewpoints and results in many of the chapters. For each topic the theoretical framework is carefully explained using examples and applications.

The second chapter on basic iterated function systems theory is designed to be used as the basis for a course and includes many exercises. This chapter, along with the three background appendices on topological and metric spaces, measure theory, and basic results from set-valued analysis, make the book suitable for self-study or as a source book for a graduate course. The other chapters illustrate many extensions and applications of fractal-based methods to different areas. This book is intended for graduate students and researchers in applied mathematics, engineering and social sciences.

Herb Kunze is a professor of mathematics at the University of Guelph in Ontario. Davide La Torre is an associate professor of mathematics in the Department of Economics, Management and Quantitative Methods of the University of Milan. Franklin Mendivil is a professor of mathematics at Acadia University in Nova Scotia. Edward Vrscay is a professor in the department of Applied Mathematics at the University of Waterloo in Ontario. The major focus of their research is on fractals and the applications of fractals.

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### Contents

 Chapter 1 What do we mean by FractalBased Analysis? 1 Chapter 2 Basic IFS Theory 21 Chapter 3 IFS on Spaces of Functions 87 Chapter 4 Iterated Function Systems Multifunctions and MeasureValued Functions 125 Chapter 5 IFS on Spaces of Measures 149 Chapter 6 The Chaos Game 213 Chapter 7 Inverse Problems and FractalBased Methods 243
 Chapter 8 Further Developments and Extensions 315 Appendix A Topological and Metric Spaces 357 Appendix B Basic Measure Theory 369 Appendix C Basic Notions from SetValued Analysis 385 References 391 Index 403 Copyright

### About the author (2011)

Franklin Mendivil is a Professor in the Department of Mathematics and Statistics at Acadia University; Herb Kunze is a Professor in the Department of Mathematics and Statistics at Guelph University; Davide La Torre is an Associate Professor in the Department of Economics, Business and Statistics at University of Milan; Edward R. Vrscay is a Professor at the University of Waterloo.