This work combines several important areas of mathematics - dynamical systems, measure theory and ergodic theory - and gives a thorough treatment of one-dimensional chaotic systems. It includes many examples and applications, and provides problem sets with solutions.
Laws of Chaos
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Editorial Reviews
Review
"Well written with ample examples and exercises… It fills an important gap in the literature." —Journal of the Indian Inst. of Science "The book provides a personal view on invariant measures and dynamical systems in one dimension. It is given a detailed study of the piecewise linear transformations under another spirit than that of {W. Doeblin} developed in the commemorative volume [Doeblin and modern probability. Proceedings of the Doeblin conference "50 years after Doeblin: Developments in the theory of Markov chains, Markov processes and sums of independent random variables'', Contemporary Mathematics 149 (1993; articles are reviewed individually in Zbl )]. The book contains 13 chapters. Some titles are as follows: Spectral decomposition of the Frobenius-Perron operator, Markov transformations, Compactness theorem and Approximation of invariant densities, Stability of invariant measures, The inverse problem for the Frobenius-Perron equation and others. The style of the book is clear with good didactical perspectives for those who wish to study dynamical systems in connection with measure theory and ergodic theory. Finally, the book is a valuable contribution to the topic of Dynamical Systems." —Zentralblatt Math
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4 of 4 people found the following review helpful:
5.0 out of 5 stars
Recommended for acim's lovers,
By Wael Bahsoun (Montreal, Qc Canada) - See all my reviews
This review is from: Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension (Probability and its Applications) (Hardcover)
This book starts with an introduction to ergodic theory. The rest of the book is on absolutely continuous invariat measures and their applications. This is one of the best books that studies conditions of existance of absolutely continuous invariat measures for piecewise expanding maps.
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Inside This Book
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First Sentence:
This book is concerned with discrete time dynamical systems defined by a piecewise continuous transformation of an interval into itself, such as is shown in Figure 1.1.1. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
piecewise expanding transformations, piecewise convex transformations, normalized measure space, doubly stochastic kernel, piecewise monotonic maps, piecewise monotonic transformations, invariant density function, tent transformation, continuous invariant measures, absolutely continuous invariant measure, asymptotic periodicity, piecewise linear transformations, invariant densities, communicating classes, measurable transformation, matrix induced, defining partition, weak mixing, bounded variation, partition points, measure preserving, ergodic properties, hounded variation, ergodic measures, ergodic theory Key Phrases - Capitalized Phrases (CAPs): (learn more)
Birkhoff Ergodic Theorem, Central Limit Theorem, Folklore Theorem, Poincaré Recurrence Theorem, Frobenius Perron, Radon-Nikodym Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
This book is concerned with discrete time dynamical systems defined by a piecewise continuous transformation of an interval into itself, such as is shown in Figure 1.1.1. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
piecewise expanding transformations, piecewise convex transformations, normalized measure space, doubly stochastic kernel, piecewise monotonic maps, piecewise monotonic transformations, invariant density function, tent transformation, continuous invariant measures, absolutely continuous invariant measure, asymptotic periodicity, piecewise linear transformations, invariant densities, communicating classes, measurable transformation, matrix induced, defining partition, weak mixing, bounded variation, partition points, measure preserving, ergodic properties, hounded variation, ergodic measures, ergodic theory Key Phrases - Capitalized Phrases (CAPs): (learn more)
Birkhoff Ergodic Theorem, Central Limit Theorem, Folklore Theorem, Poincaré Recurrence Theorem, Frobenius Perron, Radon-Nikodym Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 23
books:
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book
cites this book:
- Ergodic Theory (Cambridge Studies in Advanced Mathematics) by Karl E. Petersen on page 29, and Back Matter
- Ergodic Theory (Grundlehren der mathematischen Wissenschaften) by I. P. Kornfeld on page 29, and Back Matter
- Invariant Measures by John von Neumann on page 67, and page 96
- Dynamical Systems by Z. Coelho in Back Matter
- A Survey of Matrix Theory and Matrix Inequalities (Dover Books on Mathematics) by Marvin Marcus in Back Matter
See all 23 books this book cites
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