On the distribution of the angle between Oseledets spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909935604858880 |
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| author | Bochi, Jairo Lessa, Pablo |
| author_facet | Bochi, Jairo Lessa, Pablo |
| contents | We study the distribution of the angles between Oseledets subspaces and their log-integrability, focusing on dimension $2$. For random i.i.d. products of matrices, we construct examples of probability measures on $\mathrm{GL}_2(\mathbb{R})$ with finite first moment where the Oseledets angle is not log-integrable. We also show that for probability measures with finite second moment the angle is always log-integrable. We then consider general measurable $\mathrm{GL}_2(\mathbb{R})$-cocycles over an arbitrary ergodic automorphism of a non-atomic Lebesgue space, proving that no integrability condition on the matrix distribution ensures log-integrability of the angle. In fact, the joint distribution of the Oseledets spaces can be chosen arbitrarily. A similar flexibility result for bounded cocycles holds under an unavoidable technical restriction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_04612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the distribution of the angle between Oseledets spaces Bochi, Jairo Lessa, Pablo Dynamical Systems 37D25, 60G42, 60G50 We study the distribution of the angles between Oseledets subspaces and their log-integrability, focusing on dimension $2$. For random i.i.d. products of matrices, we construct examples of probability measures on $\mathrm{GL}_2(\mathbb{R})$ with finite first moment where the Oseledets angle is not log-integrable. We also show that for probability measures with finite second moment the angle is always log-integrable. We then consider general measurable $\mathrm{GL}_2(\mathbb{R})$-cocycles over an arbitrary ergodic automorphism of a non-atomic Lebesgue space, proving that no integrability condition on the matrix distribution ensures log-integrability of the angle. In fact, the joint distribution of the Oseledets spaces can be chosen arbitrarily. A similar flexibility result for bounded cocycles holds under an unavoidable technical restriction. |
| title | On the distribution of the angle between Oseledets spaces |
| topic | Dynamical Systems 37D25, 60G42, 60G50 |
| url | https://arxiv.org/abs/2503.04612 |