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Product Code:  SURV/276 
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Softcover ISBN:  9781470470463 
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Softcover ISBN:  9781470470463 
Product Code:  SURV/276 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470475505 
Product Code:  SURV/276.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470470463 
eBook ISBN:  9781470475505 
Product Code:  SURV/276.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 276; 2023; 451 ppMSC: Primary 28; Secondary 37; 42
Although there is no precise definition of a “fractal”, it is usually understood to be a set whose smaller parts, when magnified, resemble the whole. Selfsimilar and selfaffine sets are those for which this resemblance is precise and given by a contracting similitude or affine transformation. The present book is devoted to this most basic class of fractal objects.
The book contains both introductory material for beginners and more advanced topics, which continue to be the focus of active research. Among the latter are selfsimilar sets and measures with overlaps, including the muchstudied infinite Bernoulli convolutions. Selfaffine systems pose additional challenges; their study is often based on ergodic theory and dynamical systems methods. In the last twenty years there have been many breakthroughs in these fields, and our aim is to give introduction to some of them, often in the simplest nontrivial cases.
The book is intended for a wide audience of mathematicians interested in fractal geometry, including students. Parts of the book can be used for graduate and even advanced undergraduate courses.
ReadershipGraduate students and researchers interested in fractals and related mathematical structures.

Table of Contents

Chapters

Introduction

Elements of geometric measure theory

General properties of selfsimilar sets and measures

Separation properties for selfsimilar IFS

Multifractal analysis for selfsimilar measures

Transversality techniques for selfsimilar IFS

Further properties of selfsimilar IFS with overlaps

Fourieranalytic and numbertheoretic methods

Elements of ergodic theory

Selfaffine sets and measures

Diagonally selfaffine IFS

Exact dimensionality and dimension conservation

Local entropy averages and projections of selfaffine sets and measures

Nonlinear conformal iterated functions systems

Some elements of linear algebras

Some elements of measure theory

Some elements of harmonic analysis

Some acts about algebraic numbers


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Although there is no precise definition of a “fractal”, it is usually understood to be a set whose smaller parts, when magnified, resemble the whole. Selfsimilar and selfaffine sets are those for which this resemblance is precise and given by a contracting similitude or affine transformation. The present book is devoted to this most basic class of fractal objects.
The book contains both introductory material for beginners and more advanced topics, which continue to be the focus of active research. Among the latter are selfsimilar sets and measures with overlaps, including the muchstudied infinite Bernoulli convolutions. Selfaffine systems pose additional challenges; their study is often based on ergodic theory and dynamical systems methods. In the last twenty years there have been many breakthroughs in these fields, and our aim is to give introduction to some of them, often in the simplest nontrivial cases.
The book is intended for a wide audience of mathematicians interested in fractal geometry, including students. Parts of the book can be used for graduate and even advanced undergraduate courses.
Graduate students and researchers interested in fractals and related mathematical structures.

Chapters

Introduction

Elements of geometric measure theory

General properties of selfsimilar sets and measures

Separation properties for selfsimilar IFS

Multifractal analysis for selfsimilar measures

Transversality techniques for selfsimilar IFS

Further properties of selfsimilar IFS with overlaps

Fourieranalytic and numbertheoretic methods

Elements of ergodic theory

Selfaffine sets and measures

Diagonally selfaffine IFS

Exact dimensionality and dimension conservation

Local entropy averages and projections of selfaffine sets and measures

Nonlinear conformal iterated functions systems

Some elements of linear algebras

Some elements of measure theory

Some elements of harmonic analysis

Some acts about algebraic numbers