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 $k$.
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机译：我们的主要重点是双曲线表面上的封闭测地仪。对于任何固定的整数$ k $，我们对所有封闭的大测地测器集感兴趣，至少k $（但可能更多）自交叉口。其中，我们认为最小长度和调查其自交叉数量。我们证明了它们的交叉点数是Universal Linear函数以$ k $（适用于任何双曲线）的上限。此外，在尖瓣的存在下，我们得到了界限，这意味着自交叉点数表现得很渐离，因为k $ k $ k $。
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