Continuous-time extensions of discrete-time cocycles
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914263867588608 |
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| author | Chemnitz, Robin Engel, Maximilian Koltai, Péter |
| author_facet | Chemnitz, Robin Engel, Maximilian Koltai, Péter |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_07338 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuous-time extensions of discrete-time cocycles Chemnitz, Robin Engel, Maximilian Koltai, Péter Dynamical Systems 37C55, 37H15, 37C60 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. |
| title | Continuous-time extensions of discrete-time cocycles |
| topic | Dynamical Systems 37C55, 37H15, 37C60 |
| url | https://arxiv.org/abs/2305.07338 |