# Dynamical Numbers: Interplay Between Dynamical Systems and by Sergiy Kolyada, Yuri Manin, Martin Moller, Pieter Moree,

By Sergiy Kolyada, Yuri Manin, Martin Moller, Pieter Moree, Thomas Ward

Comprises the complaints of the task 'Dynamical Numbers: interaction among Dynamical structures and quantity concept' held on the Max Planck Institute for arithmetic (MPIM) in Bonn, from 1 could to 31 July, 2009, and the convention of an analogous identify, additionally held on the Max Planck Institute, from 20 to 24 July, 2009--Preface, p. vii

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Extra info for Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, A Special Program May 1-July 31, 2009, International Conference July 20-24, 2009, Max Planack I

Example text

The set of means on G forms a w∗ -closed convex subset of LU C(G)∗ and we say 18 HILLEL FURSTENBERG AND ELI GLASNER that an element of this set is m-stationary if m ∗ ρ = ρ. By the Markov-Kakutani ﬁxed point theorem the set of m-stationary means is nonempty. Let Z be the (compact Hausdorﬀ) Gelfand space corresponding to the C ∗ algebra L = LU C(G). Recall that Z can be viewed as the space of non-zero continuous C ∗ -homomorphisms of the C ∗ -algebra L into C. In particular, for each g ∈ G the evaluation map zg : F → F (g) is an element of Z.

Res. Stud. Math. 14, Bombay, 1998, pp. 105-117. 28 HILLEL FURSTENBERG AND ELI GLASNER 14. E. Glasner, Proximal ﬂows, Lecture Notes in Math. 517, Springer-Verlag, 1976. 15. E. Glasner, Quasifactors of minimal systems, Topol. Meth. in Nonlinear Anal. 16, (2000), 351-370. 16. E. Glasner, Ergodic theory via joinings, AMS, Surveys and Monographs, 101, 2003. 17. E. Glasner, B. Tsirelson and B. Weiss, The automorphism group of the Gaussian measure cannot act pointwise. Probability in mathematics. Israel J.

Proof. Let f : G → [0, 1] be a left uniformly continuous function such that f (g) = 1 for every g ∈ L /2 and f (g) = 0 for every g ∈ L . Let A be the uniformly closed subalgebra of the algebra LU C(G) of complex valued bounded left uniformly continuous functions on G generated by the orbit {fg : g ∈ G} of f , where fg (h) = f (gh). Let X be the (compact metric) Gelfand space corresponding to A. The fact that A is G-invariant implies that there is a naturally deﬁned G-action on X. Clearly the restriction map π : Z → X, where with the above notation Z is the Gelfand space corresponding to L, is a homomorphism of the corresponding dynamical systems.