By M. Popescu

Abelian different types with functions to jewelry and Modules (London Mathematical Society Monographs)

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3. 6 and its converse (iii) ⇒ (i) is immediate. Let us prove (ii) ⇒ (v). 4. Assume that ν is a convex combination tν1 + (1 − t)ν2 , where ν1 and ν2 are P -invariant Borel probability measures and where 0 < t < 1. For i = 1, 2, the measure νi is absolutely continuous with respect to ν and hence can be written as ϕi ν, where the function ϕi belongs to L1 (X, X , ν) and has integral 1. Since νi is P -invariant, one has P ∗ ϕi = ϕi . 4(b), one has P ϕi = ϕi , hence by assumption, ϕi = 1 ν-almost everywhere, that is, νi = ν, which was to be shown.

3 The Law of Large Numbers for Cocycles 43 Let us introduce a trick which reduces the study of cocycles with a unique average to the study of those which are centered. Replace G by G := G × Z, where Z acts trivially on X, replace μ by μ := μ ⊗ δ1 so that any μ-stationary probability measure is also μ -stationary, and replace σ by the cocycle σ : G × X → E given by σ ((g, n), x) = σ (g, x) − nσμ . 3 The Law of Large Cocycles Here is the Law of Large Numbers for cocycles. 9 Let G be a locally compact semigroup, X a compact metrizable Gspace, E a finite-dimensional real vector space and μ a Borel probability measure on G.

9. 14 Let ν be a Borel probability measure on X. (a) Then ν is μ-stationary if and only if β ⊗ ν is T X -invariant. (b) In this case, ν is μ-ergodic if and only if β ⊗ ν is T X -ergodic. 9. 15 There may exist a T X -invariant Borel probability measure on B × X whose image by the projection on the first factor is equal to β but which is not of the form β ⊗ ν for some μ-stationary Borel probability measure ν on X. e. the set of reduced one-sided infinite words in g ± and h± and μ be the probability measure μ = 12 (δg + δh ).

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