Continuous-time extensions of discrete-time cocycles
Robin Chemnitz, Maximilian Engel, Péter Koltai
TL;DR
This work addresses when a discrete-time $SL_d(\mathbb{R})$ cocycle over a compact smooth base can be realized as the time-1 map of a continuous-time cocycle over a suspension flow. The authors establish a sharp canonical-extension criterion: such an extension exists iff the discrete cocycle is nullhomotopic, and they provide a constructive method via a homotopy to produce the continuous-time generator that preserves Lyapunov exponents and hyperbolicity. In the (quasi-)periodic case, they show how to realize canonical extensions after suitable base modifications, with dimension-dependent remedies for $d=2$ versus $d\ge3$, and they demonstrate the robustness of the extension with respect to perturbations. As an application, they build a continuous-time, non-uniformly hyperbolic $SL_2$ cocycle over a uniquely ergodic driving, obtained as a canonical extension of a nullhomotopic discrete-time cocycle. Overall, the results clarify the relationship between discrete and continuous-time cocycles, preserve key dynamical quantities, and yield new examples in non-autonomous, uniquely ergodic settings.
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.