Repository: Freie Universität Berlin, Math Department

Continuous-time extensions of discrete-time cocycles

Chemnitz, Robin and Engel, Maximilian and Koltai, Péter (2023) Continuous-time extensions of discrete-time cocycles. Contemporary Mathematics . (In Press)

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We consider linear cocycles taking values in $\textup{SL}_d(\mathbb{R})$ driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is natural in the sense that the base is extended to an associated suspension flow and that the dimension of the cocycle does not change. Further, we refine our general result for the case of (quasi-)periodic driving. As an example, we present a discrete-time cocycle due to Michael Herman. The Furstenberg--Kesten limits of this cocycle do not exist everywhere and its Oseledets splitting is discontinuous. Our results on the continuous-time extension of discrete-time cocycles allow us to construct a continuous-time cocycle with analogous properties.

Item Type:Article
Subjects:Mathematical and Computer Sciences
Mathematical and Computer Sciences > Mathematics
Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics
ID Code:3067
Deposited By: Jana Jerosch
Deposited On:05 Feb 2024 12:58
Last Modified:05 Feb 2024 12:58

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