Abstract
The subject of this paper prepared as the report for NOMA conference (June 2017), is the asymptotic behaviour at large times of scalar (additive) and operator (multiplicative) cocycles in the case of ℝ – action with a finite invariant measure.
Highlights
Our first aim is to discuss the asymptotic behaviour of the general additive scalar 1 – cocycle taking integrable values as a vector-function of t, α(t, ·) ∈ L1(X, μ)
What type of asymptotic behaviour as t → ∞ can have cocycle α : R → L1, α → α(t, ·), in the general case of not necessarily absolute continuous dependence on t?. This question is of interest from different points of view and, primarily, in relation with the spectral theory of R – actionts by weighted shift operators and the metric theory of Liapoanov exponents for flows with an invariant measure
For any ergodic flow T = {T s}s∈R its special representation with the base space (Y, ν), a ceiling function f : Y → R+, f dν = 1 and a scalar additive cocycle α associated with
Summary
Our first aim is to discuss the asymptotic behaviour of the general additive scalar 1 – cocycle taking integrable values as a vector-function of t, α(t, ·) ∈ L1(X, μ). Writes: It is unknown to me, whether there exists a measurable additive cocycle α with integrable values at every t such that the statement as in the Birkhoff Theorem [concerning convergence of averages t−1α(t, x)] fails.
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