On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups

Abstract

On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups

Distal systems in topological dynamics and ergodic theory

“…If has no atoms, then [Der00, Theorem 1.1] shows the existence of a Borel measurable map satisfying (ii). However, an inspection of the proof (see the remarks after [Der00, Theorem 1.2]) reveals that for a given one can also ensure (i).…”

“…The proof rests on the following approximation result which is, in essence, due to Derrien (see [Der00]). It shows that, given a measurable map with values in the compact group U(n) of unitary n × n matrices, one can find an arbitrarily close continuous map that is cohomologous (cf.…”

“…The key step is to show that an isometric extension of measure-preserving systems admits a completely canonical topological model that is a (pseudo)isometric extension of topological dynamical systems. The proof of this requires a new functional-analytic characterization of structured extensions in topological dynamics as established in [EK21], related results from ergodic theory (see [EHK21]), as well as a result of Derrien on approximation of measurable cocycles by continuous ones (see [Der00]). The functional-analytic view makes the parallels between the structure theory of topological and measure-preserving dynamical systems more apparent and leads to canonical models in a straightforward way.…”

Distal systems in topological dynamics and ergodic theory

“…The key step is to show that an isometric extension of measure-preserving systems admits a completely canonical topological model that is a (pseudo)isometric extension of topological dynamical systems. The proof of this requires a new functional-analytic characterization of structured extensions in topological dynamics as established in [EK21], related results from ergodic theory (see [EHK21]), as well as a result of Derrien on approximation of measurable cocycles by continuous ones (see [Der00]). The functional-analytic view makes the parallels between the structure theory of topological and measure-preserving dynamical systems more apparent and leads to canonical models in a straightforward way.…”

“…The proof rests on the following approximation result which is, in essence, due to Derrien (see [Der00]). It shows that, given a measurable map with values in the compact group U( ) of unitary × -matrices, one can find an arbitrarily close continuous map that is cohomologous.…”