A dichotomy for higher dimensional flows

TL;DR

This paper investigates the fundamental division in higher-dimensional flows between those satisfying sectional-Axiom A conditions and those with points accumulated by periodic orbits of varying indices, specifically for generic flows with certain singularity properties.

Contribution

It proves a dichotomy for $C^1$ generic flows with singularities of codimension one, improving previous results in the field.

Findings

Established a dichotomy for higher-dimensional flows.

Extended previous results to a broader class of flows.

Provided new insights into the structure of flows with accumulated periodic orbits.

Abstract

We analyze the dichotomy between {\em sectional-Axiom A flows} (c.f. \cite{memo}) and flows with points accumulated by periodic orbits of different indices. Indeed, this is proved for $C^1$ generic flows whose singularities accumulated by periodic orbits have codimension one. Our result improves \cite{mp1}.