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Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms

Karl Winsor

Abstract

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.

Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms

Abstract

Let be a connected component of a stratum of the moduli space of holomorphic -forms of genus . We show that the absolute period foliation of is ergodic on the area- locus, and that the non-dense leaves lie in an explicit countable union of suborbifolds, subject to a mild assumption on . We show similar results for subspaces of defined by topological restrictions on the absolute periods. We obtain these dynamical results by showing that for a typical positive cohomology class in , the associated space of isoperiodic forms in is connected. Lastly, we show that certain covering constructions provide examples of spaces of isoperiodic forms with positive dimension and infinitely many connected components.

Paper Structure

This paper contains 67 sections, 59 theorems, 167 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Absolute periods.
  3. Main results: dynamics and topology.
  4. Disconnected spaces of isoperiodic forms.
  5. Open questions.
  6. Methods.
  7. Notes and references.
  8. Acknowledgements.
  9. Splitting zeros and connected sums
  10. Holomorphic $1$-forms.
  11. Strata.
  12. Period coordinates.
  13. Area.
  14. The $\mathop{\mathrm{GL}}\nolimits^+(2,\mathbb{R})$-action.
  15. Connected components of strata.
  16. ...and 52 more sections

Key Result

Theorem 1.1

Let $\phi \in H^1(S_g;\mathbb{C})$ be a positive cohomology class such that $\mathop{\mathrm{Per}}\nolimits(\phi) \cong \mathbb{Z}^{2g}$ and $\mathop{\mathrm{Per}}\nolimits(\phi) \cap \mathbb{R} z \subset \mathbb{Q} z$ for all $z \in \mathop{\mathrm{Per}}\nolimits(\phi)$. Then $\mathcal{C}(\phi)$ is

Figures (6)

  • Figure 1: A holomorphic $1$-form in $\Omega\mathcal{M}_2(1,1)$ (right) that arises from a holomorphic $1$-form in $\Omega\mathcal{M}_2(2)$ (left) by splitting a zero. The two segments being slit are shown in bold.
  • Figure 2: A holomorphic $1$-form $(X,\omega) \in \Omega\mathcal{M}_3(3,1)$ arising from a holomorphic $1$-form in $\Omega\mathcal{M}_2(1,1)$ by a connected sum with a torus. The pair of rightward homologous saddle connections $\alpha^{\pm}$ is a splitting of $(X,\omega)$.
  • Figure 3: A holomorphic $1$-form in the intersection of $A((3,3),(3,1),(3,1))$ with the nonhyperelliptic component of $\Omega\mathcal{M}_4(3,3)$.
  • Figure 4: A loop $\widetilde{s} : \mathbb{R} \rightarrow \widetilde{L}$ around a point in the metric completion of a leaf $\widetilde{L}$ of $\mathcal{A}(\kappa;m)$, where $\kappa = (3,1)$, $m = 3$, and $\widetilde{s}(\mathbb{R}) \subset \widetilde{A}((3,1),(3,1),(1,1))$. Along $\widetilde{s}(\mathbb{R})$, the saddle connections $\gamma_1,\gamma_2$ are rotated counterclockwise around the zero of order $3$. The $4$ images show $\widetilde{s}(0)$ (top-left), $\widetilde{s}(2\pi)$ (top-right), $\widetilde{s}(4\pi)$ (bottom-left), and $\widetilde{s}(6\pi)$ (bottom-right). We have $s(0) = s(6 \pi)$, but the prongs (shown with dashes) are different. In this case, the $4$ possible choices of prongs are realized in $p^{-1}(s(0)) = \{\widetilde{s}(0),\widetilde{s}(6\pi),\widetilde{s}(12\pi),\widetilde{s}(18\pi)\}$, and $\widetilde{s}(24\pi) = \widetilde{s}(0)$.
  • Figure 5: Left: a holomorphic $1$-form in $\Omega\mathcal{M}_3(3,1)$ with a splitting $\alpha^\pm$ and a second splitting $\gamma^\pm$ as in Lemma \ref{['lem:newsum']}. Right: the parallelogram $P$ and the cylinder $C$, with the parallelogram sides $\alpha^\pm$ and $\gamma_0^\pm$ labelled.
  • ...and 1 more figures

Theorems & Definitions (111)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • ...and 101 more