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.

中文翻译：

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具有自相交的短闭合测地线

我们的主要关注点是双曲曲面上的一组封闭测地线。对于任何固定整数ķ，我们对所有闭合测地线的集合感兴趣，至少ķ（但可能更多）自相交。其中，我们考虑那些最小长度的并研究它们的自交数。我们证明了它们的交点数是由一个通用线性函数在ķ（适用于任何双曲曲面）。此外，在存在尖点的情况下，我们得到的界限意味着自交数的行为渐近ķ为了成长ķ.