# Counting one sided simple closed geodesics on Fuchsian thrice punctured projective planes

## TL;DR

This paper establishes an asymptotic counting formula for one-sided simple closed geodesics on Fuchsian thrice punctured projective planes, revealing a noninteger growth exponent independent of hyperbolic structure.

## Contribution

It provides the first true asymptotic formula for counting one-sided simple closed geodesics on these non-orientable surfaces, with a novel noninteger growth exponent.

## Key findings

- Asymptotic formula for geodesic count established
- Growth exponent is noninteger and structure-independent
- Contrasts with Mirzakhani's results for orientable surfaces

## Abstract

We prove that there is a true asymptotic formula for the number of one sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic structure, and it is noninteger, in contrast to counting results of Mirzakhani for simple closed curves on orientable Fuchsian surfaces.