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9781470470463 Academic Inspection Copy

Self-similar and Self-affine Sets and Measures

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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. 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. 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.
Balazs Barany, Budapest University of Technology and Economics, Hungary. Karoly Simon, Budapest University of Technology and Economics, Hungary. Boris Solomyak, Bar-Ilan University, Ramat Gan, Israel.
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 Transversality 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 Some elements of linear algebras Some elements of measure theory Some elements of harmonic analysis Some acts about algebraic numbers Bibliography Index Basic notation
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