Gutierrez-Sotomayor Flows on Singular Surfaces

TL;DR

This paper investigates the realization of Lyapunov graphs with GS singularities as continuous flows on singular surfaces and computes the Euler characteristic based on singularity types.

Contribution

It provides a classification of minimal isolating blocks for GS singularities and relates singularity types to the Euler characteristic of the surface.

Findings

Complete classification of minimal isolating blocks for GS singularities

Euler characteristic expressed in terms of singularity types

Conditions for realizing Lyapunov graphs with GS singularities as flows

Abstract

In this work we address the realizability of a Lyapunov graph labeled with GS singularities, namely regular, cone, Whitney, double crossing and triple crossing singularities, as continuous flow on a singular closed $2$-manifold $\mathbf{M}$. Furthermore, the Euler characteristic is computed with respect to the types of GS singularities of the flow on $\mathbf{M}$. Locally, a complete classification theorem for minimal isolating blocks of GS singularities is presented in terms of the branched one manifolds that make up the boundary.