Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866929505008877568 |
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| author | Winsor, Karl |
| author_facet | Winsor, Karl |
| contents | Let $\mathcal{C}$ be a connected component of a stratum of the moduli space of holomorphic $1$-forms of genus $g$. We show that the absolute period foliation of $\mathcal{C}$ is ergodic on the area-$1$ locus, and that the non-dense leaves lie in an explicit countable union of suborbifolds, subject to a mild assumption on $\mathcal{C}$. We show similar results for subspaces of $\mathcal{C}$ defined by topological restrictions on the absolute periods. We obtain these dynamical results by showing that for a typical positive cohomology class in $H^1(S_g;\mathbb{C})$, the associated space of isoperiodic forms in $\mathcal{C}$ is connected. Lastly, we show that certain covering constructions provide examples of spaces of isoperiodic forms with positive dimension and infinitely many connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_12669 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms Winsor, Karl Dynamical Systems Complex Variables Geometric Topology Let $\mathcal{C}$ be a connected component of a stratum of the moduli space of holomorphic $1$-forms of genus $g$. We show that the absolute period foliation of $\mathcal{C}$ is ergodic on the area-$1$ locus, and that the non-dense leaves lie in an explicit countable union of suborbifolds, subject to a mild assumption on $\mathcal{C}$. We show similar results for subspaces of $\mathcal{C}$ defined by topological restrictions on the absolute periods. We obtain these dynamical results by showing that for a typical positive cohomology class in $H^1(S_g;\mathbb{C})$, the associated space of isoperiodic forms in $\mathcal{C}$ is connected. Lastly, we show that certain covering constructions provide examples of spaces of isoperiodic forms with positive dimension and infinitely many connected components. |
| title | Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms |
| topic | Dynamical Systems Complex Variables Geometric Topology |
| url | https://arxiv.org/abs/2109.12669 |