Dispersing cocycles and mixing flows under functions

Klaus Schmidt. "Dispersing cocycles and mixing flows under functions." Fundamenta Mathematicae 173.2 (2002): 191-199. <http://eudml.org/doc/283317>.

@article{KlausSchmidt2002,
abstract = {Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X,,μ) and A a locally compact second countable abelian group. A cocycle c: G × X → A for T disperses if $lim_\{g→∞\}c(g,·) - α(g) = ∞$ 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 ergodic automorphism of a probability space (X,,μ), f: X → ℝ be a nonnegative Borel map with ∫fdμ = 1, and $T^f$ 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].},
author = {Klaus Schmidt},
journal = {Fundamenta Mathematicae},
keywords = {boundedness; tightness and dispersion of cocycles; mixing properties of flows},
language = {eng},
number = {2},
pages = {191-199},
title = {Dispersing cocycles and mixing flows under functions},
url = {http://eudml.org/doc/283317},
volume = {173},
year = {2002},
}

TY - JOUR
AU - Klaus Schmidt
TI - Dispersing cocycles and mixing flows under functions
JO - Fundamenta Mathematicae
PY - 2002
VL - 173
IS - 2
SP - 191
EP - 199
AB - Let T be a measure-preserving and mixing action of a countable abelian group G on a probability space (X,,μ) and A a locally compact second countable abelian group. A cocycle c: G × X → A for T disperses if $lim_{g→∞}c(g,·) - α(g) = ∞$ 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 ergodic automorphism of a probability space (X,,μ), f: X → ℝ be a nonnegative Borel map with ∫fdμ = 1, and $T^f$ 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].
LA - eng
KW - boundedness; tightness and dispersion of cocycles; mixing properties of flows
UR - http://eudml.org/doc/283317
ER -