Counting invariant components of hyperelliptic translation surfaces

2013/02/14 by Kathryn Lindsey, Lindsey, Kathryn
Mathematics · #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1302.3282

openalex publication_date 2013/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The flow in a fixed direction on a translation surface S determines a decomposition of S into closed invariant sets, each of which is either periodic or minimal. We study this decomposition for translation surfaces in the hyperelliptic connected components H^hyp(2g-2) and H^hyp(g-1,g-1) of the corresponding strata of the moduli space of translation surfaces. Specifically, we characterize the pairs of nonnegative integers (p,m) for which there exists a translation surface in H^hyp(2g-2) or H^hyp(g-1,g-1) with precisely p periodic components and m minimal components. This extends results by Naveh ([Naveh08]), who obtained tight upper bounds on the numbers of minimal components and invariant components a translation surface in any given stratum may have. Analogous results for the other connected components of moduli space are forthcoming.

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