Reference : Short closed geodesics with self-intersections
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Physical, chemical, mathematical & earth Sciences : Mathematics
Short closed geodesics with self-intersections
Erlandsson, Viveka [> >]
Parlier, Hugo mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit]
[en] Mathematics - Geometric Topology ; Mathematics - Differential Geometry
[en] Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in $k$ (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like $k$ for growing $k$.
19 pages, 5 figures

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