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arXiv · 2305.07338

Continuous-time extensions of discrete-time cocycles

Robin Chemnitz·Maximilian Engel·Péter Koltai
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Abstract

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 canonical in the sense that the base is extended to an associated suspension flow and that the discrete-time cocycle is recovered as the time-1 map of the continuous-time cocycle. Further, we refine our general result for the case of (quasi-)periodic driving. We use our findings to construct a non-uniformly hyperbolic continuous-time cocycle in $\SL{2}$ over a uniquely ergodic driving.

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Robin Chemnitz, Maximilian Engel, Péter Koltai. 2026-01-20. Continuous-time extensions of discrete-time cocycles. https://arxiv.org/abs/2305.07338

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