Bounds on the number of non-simple closed geodesics on a surface

TL;DR

This paper establishes tighter exponential bounds on the number of non-simple closed geodesics on negatively curved surfaces, based on length and self-intersection constraints, refining previous growth estimates.

Contribution

It provides new exponential bounds on non-simple closed geodesics considering both length and self-intersection limits, improving prior results.

Findings

Number of non-simple closed geodesics grows exponentially with length.

Bounds depend on both length and self-intersection constraints.

Results tighten previous growth estimates under weak conditions.

Abstract

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given weak conditions on self-intersection number.