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Invitation to Ergodic Theory
C. E. Silva, Williams College, Williamstown, MA
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Student Mathematical Library
2008; 262 pp; softcover
Volume: 42
ISBN-10: 0-8218-4420-2
ISBN-13: 978-0-8218-4420-5
List Price: US$46 Institutional Members: US$36.80
All Individuals: US\$36.80
Order Code: STML/42

Lectures on Fractal Geometry and Dynamical Systems - Yakov Pesin and Vaughn Climenhaga

Ergodic Theory, Groups, and Geometry - Robert J Zimmer and Dave Witte Morris

Mirror Symmetry and Algebraic Geometry - David A Cox and Sheldon Katz

This book is an introduction to basic concepts in ergodic theory such as recurrence, ergodicity, the ergodic theorem, mixing, and weak mixing. It does not assume knowledge of measure theory; all the results needed from measure theory are presented from scratch. In particular, the book includes a detailed construction of the Lebesgue measure on the real line and an introduction to measure spaces up to the Carathéodory extension theorem. It also develops the Lebesgue theory of integration, including the dominated convergence theorem and an introduction to the Lebesgue $$L^p$$spaces.

Several examples of a dynamical system are developed in detail to illustrate various dynamical concepts. These include in particular the baker's transformation, irrational rotations, the dyadic odometer, the Hajian-Kakutani transformation, the Gauss transformation, and the Chacón transformation. There is a detailed discussion of cutting and stacking transformations in ergodic theory. The book includes several exercises and some open questions to give the flavor of current research. The book also introduces some notions from topological dynamics, such as minimality, transitivity and symbolic spaces; and develops some metric topology, including the Baire category theorem.

Reviews

"The writing is crisp and clear. Proofs are written carefully with adequate levels of detail. Exercises are plentiful and well-integrated with the text."

-- MAA Reviews

"The author presents in a very pleasant and readable way an introduction to ergodic theory for measure-preserving transformations of probability spaces. In my opinion, the book provides guidelines, classical examples and useful ideas for an introductory course in ergodic theory to students that have not necessarily already been taught Lebesgue measure theory."

-- Elemente der Mathematik

"The book contains many (often easy or very easy) exercises, both in the text as well as at the end of each section."

-- Mathematical Reviews