A genus 4 origami with minimal hitting time and an intersection property

TL;DR

This paper investigates the hitting time in minimal flows on genus 4 origamis, establishing a link between hitting time and Diophantine type for the Ornithorynque origami and providing a general criterion for this relationship.

Contribution

It introduces a new connection between hitting time and Diophantine type on a specific genus 4 origami and offers a general criterion for such equality in minimal flows.

Findings

Hitting time equals Diophantine type on Ornithorynque origami.

Hitting time generally exceeds Diophantine type for genus ≥ 2.

Provides a criterion for equality between hitting time and Diophantine type.

Abstract

In a minimal flow, the hitting time is the exponent of the power law, as r goes to zero, for the time needed by orbits to become r-dense. We show that on the so-called Ornithorynque origami the hitting time of the flow in an irrational slope equals the diophantine type of the slope. We give a general criterion for such equality. In general, for genus at least two, hitting time is strictly bigger than diophantine type.