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Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $$(M, \xi , \mathcal {F})$$ whose convex boundary is equipped with a signed singular foliation $$\mathcal {F}$$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $$c_D$$ and $$c_A$$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $$\mathcal {F}$$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić.
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Effective equidistribution of large dimensional measures on affine invariant submanifolds
The unstable foliation, that locally is given by changing horizontal components of period coordinates, plays an important role in study of translation surfaces, including their deformation theory and in the understanding of horocycle invariant measures. In this article we show that measures of large dimension on the unstable foliation equidistribute in affine invariant submanifolds and give an effective rate. An analogous result in the setting of homogeneous dynamics is crucially used in the effective density and equidistribution results of Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang.
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- Award ID(s):
- 2103136
- PAR ID:
- 10639736
- Publisher / Repository:
- arXiv
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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