Expansive Flows of the Three-Sphere

TL;DR

This paper demonstrates that the three-sphere can support transitive expansive flows with hyperbolic equilibrium points, constructed via a geodesic flow on a hyperbolic three-punctured sphere embedded in the three-sphere.

Contribution

It introduces a novel construction linking geodesic flows on hyperbolic surfaces to expansive flows on the three-sphere, expanding understanding of dynamical systems on manifolds.

Findings

Existence of transitive expansive flows on the three-sphere.

Construction method using geodesic flow of hyperbolic three-punctured sphere.

Flow exhibits hyperbolic equilibrium points.

Abstract

In this article we show that the three-dimensional sphere admits {transitive} expansive flows in the sense of Komuro with hyperbolic equilibrium points. The result is based on a construction that allows us to see the geodesic flow of a hyperbolic three-punctured two-dimensional sphere as the flow of a smooth vector field on the three-dimensional sphere.