Gutierrez-Sotomayor flows on singular surfaces - de Rezende, Ketty A.; Grulha Jr., Nivaldo G.; Lima, Dahisy V. de S.; Zigart, Murilo A.J. (Topological Methods in Nonlinear Analysis, 2022)
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 R, cone C, Whitney W, double D and triple 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 R, C, W, D and 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 R, C and W singularities or that it be linear.