Star vector fields on three-manifolds are multi-singular hyperbolic

TL;DR

This paper proves that all three-dimensional star flows are multi-singular hyperbolic, extending prior results and providing a unified framework for understanding hyperbolic structures in vector fields with singularities.

Contribution

It generalizes previous characterizations by showing that vector fields with all periodic orbits of the same index are multi-singular hyperbolic, especially in three dimensions.

Findings

All three-dimensional star flows are multi-singular hyperbolic.

Star flows with singularities of the same index are singular hyperbolic.

Star flows with robust chain recurrence classes are multi-singular hyperbolic.

Abstract

The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits robustly hyperbolic (star flows), are hyperbolic in absence of singularities. Morales, Pacifico and Pujals proposed a partial hyperbolicity called "singular hyperbolicity" that characterizes an open and dense subset of three dimensional star flows. In higher dimensions, Bonatti and da Luz characterize an open and dense set of star vector fields by multi-singular hyperbolicity. In this article, we prove that a vector field exhibiting all periodic orbits robustly of the same index is multi-singular hyperbolic, generalizing the previous results. As a corollary, we obtained that all three-dimensional star flows are multi-singular hyperbolic. Moreover, if all singularities in the same class exhibit the same index, the star flow is singular hyperbolic. Additionally, star flows with robust chain recurrence classes in any dimension are multisingular hyperbolic.