# Operator Theoretic Aspects of Ergodic Theory (Graduate Texts in Mathematics)

Language: English

Pages: 628

ISBN: 3319168975

Format: PDF / Kindle (mobi) / ePub

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unitary system for (π, E) if the vectors are linearly independent, is π G -invariant and the corresponding matrix representation defined by is unitary, i.e., satisfies for all x ∈ G. A unitary system is called irreducible if F does not contain any nontrivial π G -invariant subspaces. Lemma 15.13. Let be a continuous representation of the compact group G on some Banach space E. a) Let be a finite-dimensional unitary representation of G and u ∈ E. Then the finite-dimensional space is π G

Proposition 2.33. Let be a topological system and consider the following assertions: (i) is forward transitive, i.e., there is a point x ∈ K with . (ii) For all open sets U,V ≠ ∅ in K there is with . (iii) For all open sets U,V ≠ ∅ in K there is with . Then (ii) and (iii) are equivalent, (ii) implies (i) if K is metrizable, and (i) implies (ii) if K has no isolated points. Proof. The proof of the equivalence of (ii) and (iii) is left to the reader. (i)(ii): Suppose that K has no

Lemma 19.32. Let K be a compact space, let be a commutative semigroup of self-homeomorphisms of K acting minimally on K, let and let be an IP set. Suppose that for every , for every finite open covering there is and n ∈ I such that the intersection in (19.1) is nonempty. Then for every open set we have Proof. Let be an open set. Since is closed and invariant under the action of , it must be empty by minimality of the action. By compactness there are such that . By the assumption there is }

(German). M. Einsiedler and T. Ward [2011] Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag, London, 2011. [1943] Induced measure preserving transformations, Proc. Imp. Acad., Tokyo 19 (1943), 635–641. K. Petersen [1989] Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1989. Corrected reprint of the 1983 original. V. A. Rokhlin [1948] A “general” measure-preserving transformation

[1979] Theorie der Gleichverteilung, Bibliographisches Institut, Mannheim, 1979. L. Kuipers and H. Niederreiter [1974] Uniform Distribution of Sequences, Pure and Applied Mathematics, Interscience Publishers [John Wiley & Sons, Inc.], New York, 1974. R. R. Phelps [1966] Lectures on Choquet’s Theorem, D. Van Nostrand Co., Inc., Princeton, NJ-Toronto, ON-London, 1966. R. A. Raimi [1964] Minimal sets and ergodic measures in βN − N, Bull. Amer. Math. Soc. 70 (1964), 711–712.MATHMathSciNetCrossRef