Gutierrez-Sotomayor flows on singular surfaces

Abstract

In this work, we consider the collection of necessary homological conditions previously obtained via Conley index theory for a Lyapunov semi-graph to be associated to a Gutierrez-Sotomayor flow on an isolating block and address their sufficiency. These singular flows include regular $\mathcal{R}$, cone $\mathcal{C}$, Whitney $\mathcal{W}$, double $\mathcal{D}$ and triple $\mathcal{T}$ crossing singularities. Local sufficiency of these conditions are proved in the case of Lyapunov semi-graphs along with a complete characterization of the branched $1$-manifolds that make up the boundary of the block. As a consequence, global sufficient conditions are determined for Lyapunov graphs labelled with $\mathcal{R}$, $\mathcal{C}$, $\mathcal{W}$, $\mathcal{D}$ and $\mathcal{T}$ and with minimal weights to be associated to Gutierrez-Sotomayor flows on closed singular $2$-manifolds. By removing the minimality condition, we prove other global realizability results by requiring that the Lyapunov graph be labelled with $\mathcal{R}$, $\mathcal{C}$ and $\mathcal{W}$ singularities or that it be linear.