Wada Boundaries on a Hyperbolic Pair of Pants

TL;DR

This paper investigates the complex behavior of geodesics on a hyperbolic pair of pants, revealing Wada basin boundaries and how escape channels influence system entropy through numerical analysis.

Contribution

It introduces a numerical approach to analyze geodesic dynamics on a hyperbolic pair of pants, demonstrating Wada basin boundaries and the dependence of basin entropy on channel opening angles.

Findings

Most geodesics escape through distinct channels

Basin boundaries are of the Wada type

Basin entropy varies with channel opening angle

Abstract

In this paper the geodesics of an open multiply connected hyperbolic manifold are presented from the dynamical system point of view. The approach is completely numerical. Similar to the closed hyperbolic case there is a zero-measure set of periodic orbits. The difference is that now, most of the geodesics escape to infinity through one of the topologically distinct channels, and initial conditions are identified to which channel the orbit escapes. The initial condition mesh reveals basins of attraction to each channel. We verified that the basin boundary points are of the Wada type by two independent methods. We have also calculated the basin entropy of the system, and verified that it depends on the opening angle of the escape channels. The manifold chosen is the pair of pants together with the leaves, so there are $3$ distinct exit channels.