Dispersing Cocycles and Mixing Flows under Functions

نویسنده

  • KLAUS SCHMIDT
چکیده

Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X, S, μ) and A a locally compact second countable abelian group. A cocycle c : G × X 7−→ A for T disperses if limg→∞ c(g, ·) − α(g) = ∞ in measure for every map α : G −→ A. We prove that such a cocycle c does not disperse if and only if there exists a compact subgroup A0 ⊂ A such that the composition θ ◦ c : G × X −→ A/A0 of c with the quotient map θ : A −→ A/A0 is trivial (i.e. cohomologous to a homomorphism η : G −→ A/A0). This result extends a number of earlier characterizations of coboundaries and trivial cocycles by tightness conditions on the distributions of the maps {c(g, ·) : g ∈ G} and has implications for flows under functions: let T be a measure-preserving and ergodic automorphism of a probability space (X, S, μ), f : X −→ R a positive Borel map with R f dμ = 1, and let T f be the flow under the function f with base T . Our main result implies that, if T is mixing and T f is weakly mixing, or if T is ergodic and T f is mixing, then the cocycle f : Z ×X −→ R defined by f disperses. The latter statement answers a question raised by Mariusz Lemańczyk in [7]. 1. Dispersion of cocycles Definition 1.1. Let T : g 7→ Tg be a measure-preserving action of a countable additive abelian group G on a standard probability space (X, S, μ), and let A be a locally compact second countable additive abelian group with identity element 0. A Borel map c : G×X −→ A is a cocycle for T if c(g, Thx) + c(h, x) = c(g + h, x) for every g, h ∈ G and x ∈ X. Two cocycles c, c′ : G×X −→ A are cohomologous if there exists a Borel map b : X −→ A such that c(g, x) = c′(g, x) + b(Tgx)− b(x) (1.1) for every g ∈ G and μ-a.e. x ∈ X. The map b in (1.1) is called a transfer function. If c is cohomologous to the zero cocycle c′ ≡ 0 then c is a coboundary with transfer (or cobounding) function b. Let c : G × X −→ A be a cocycle. The cocycle c is a homomorphism if the map c(g, ·) : X −→ A is constant for every g ∈ G, and c is trivial if it is cohomologous to a homomorphism. 2000 Mathematics Subject Classification. 37A05, 37A10, 37A20, 37A25, 37H05.

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تاریخ انتشار 2002