This classic book is based on lectures given by the author at the University of Chicago in 1956. The topics covered include, in particular, recurrence, the ergodic theorems, and a general discussion of ergodicity and mixing properties. There is also a general discussion of the relation between conjugacy and equivalence. With minimal prerequisites of some analysis and measure theory, this work can be used for a one-semester course in ergodic theory or for self-study. Readership Graduate students and research mathematicians interested in number theory. Table of Contents Introduction Examples Recurrence Mean convergence Pointwise convergence Comments on the ergodic theorem Ergodicity Consequences of ergodicity Mixing Measure algebras Discrete spectrum Automorphisms of compact groups Generalized proper values Weak topology Weak approximation Uniform topology Uniform approximation Category Invariant measures Invariant measures: the solution Invariant measures: the problem Generalized ergodic theorems Unsolved problems References
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A Clear Exposition of Ergodic Theory,
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This review is from: Lectures on Ergodic Theory (AMS Chelsea Publishing) (Hardcover)
I am a retired mathematician and was curious as to what ergodic theory was about. I bought several books on the subject and found Halmos' book by far the the most helpful. He leads one into the subject in a straightforward manner avoiding the use of a tiresome complicated system of terminology and symbols. He only assumes a knowledge of analysis in the reader and in other parts of mathematics either enunciates the result to be used or gives a sufficient reference to it. His exposition is succinct and well phrased, and he gives many examples to illustrate the conceptions he introduces.
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First Sentence:
The basic concept is that of a measure space, i.e., a set X together with a specified sigma-algebra of subsets of X and a measure defined on that algebra. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
random ergodic theorem, unitary operator induced, second category theorem, discrete spectrum theorem, maximal ergodic theorem, dyadic sets, weak approximation theorem, ergodic transformations, mixing transformations, bilateral shift, recurrence theorem, mean ergodic theorem, uniform topology, finite orbits, first subinterval, measure algebra, constant absolute value, proper vector, dyadic interval, measurable transformation, finite measure, weak topology, countable base Key Phrases - Capitalized Phrases (CAPs): (learn more)
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The basic concept is that of a measure space, i.e., a set X together with a specified sigma-algebra of subsets of X and a measure defined on that algebra. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
random ergodic theorem, unitary operator induced, second category theorem, discrete spectrum theorem, maximal ergodic theorem, dyadic sets, weak approximation theorem, ergodic transformations, mixing transformations, bilateral shift, recurrence theorem, mean ergodic theorem, uniform topology, finite orbits, first subinterval, measure algebra, constant absolute value, proper vector, dyadic interval, measurable transformation, finite measure, weak topology, countable base Key Phrases - Capitalized Phrases (CAPs): (learn more)
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Citations (learn more)
45
books
cite this book:
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- Ergodic Theory and Harmonic Analysis: Proceedings of the 1993 Alexandria Conference (London Mathematical Society Lecture Note Series) by Karl E. Petersen on page 149, and page 245
- Descriptive Set Theory and Dynamical Systems (London Mathematical Society Lecture Note Series) by M. Foreman on page 28, and page 56
- Function Spaces: The Fifth Conference (Lecture Notes in Pure and Applied Mathematics) by Henryk Hudzik on page 179
- Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon (University Lecture Series) by Vladimir Pestov in Back Matter
- Easy as Pi?: An Introduction to Higher Mathematics by Oleg A. Ivanov in Back Matter
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