A Birman-Series type result for geodesics with infinitely many self-intersections

Sapir, Jenya

doi:10.1090/tran/8108

A Birman-Series type result for geodesics with infinitely many self-intersections
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by Jenya Sapir PDF

Trans. Amer. Math. Soc. 374 (2021), 7553-7568

Abstract:

Given a hyperbolic surface $\mathcal {S}$, a classic result of Birman and Series states that for each $K$, all complete geodesics with at most $K$ self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of $\mathcal {S}$. We define a self-intersection function for each complete geodesic, which bounds the number of self-intersections in finite length subarcs. We then extend the Birman-Series result to sets of complete geodesics with certain bounds on their self-intersection functions. In fact, we get the same conclusion as the Birman-Series result for sets of complete geodesics whose self-intersection functions are in $o(l^2)$, where $l$ measures arclength.

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