Self-similar and self-affine sets and measures / Balázs Bárány, Károly Simon, Boris Solomyak.

Format:

Book

Author/Creator:

Bárány, Balázs, 1984- Author.

Simon, Károly, 1961- Author.

Solomyak, Boris, 1959- Author.

Series:

Mathematical surveys and monographs ; no. 276.

Mathematical surveys and monographs, 0076-5376 ; Volume 276

Language:

English

Subjects (All):

Self-similar processes.

Fractals.

Hausdorff measures.

fractals.

Genre:

Informational works.

Physical Description:

xii, 451 pages : illustrations (some color) ; 26 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2023]

Summary:

"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. Self-similar and self-affine 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 self-similar sets and measures with overlaps, including the much-studied infinite Bernoulli convolutions. Self-affine 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"-- Publisher's website.

Contents:

Introduction

Elements of geometric measure theory

General properties of self-similar sets and measures

Separation properties for self-similar IFS

Multifractal analysis for self-similar measures

Trasversality techniques for self-similar IFS

Further properties of self-similar IFS with overlaps

Fourier-analytic and number-theoretic methods

Elements of Ergodic Theory

Self-affine sets and measures

Diagonally self-affine IFS

Exact dimensionality and dimension conservation

Local entropy averages and projections of self-affine sets and measures

Nonlinear conformal iterated functions systems

Appendices. Some elements of linear algebra ; Some elements of measure theory ; Some elements of Harmonic Analysis ; Some facts about algebraic numbers.

Notes:

Includes bibliographical references (pages 429-445) and index.

ISBN:

9781470470463

1470470462

OCLC:

1401628230