Gutierrez-Sotomayor flows: isolating blocks and homotopical cancellations

Autor(es)

de Rezende, Ketty A. ; Lima, Dahisy V. S. ; Zigart, Murilo A. de Jesus

• Tipo de documento: Artigo Científico
• Fonte: SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES; v. 19, n. 1, p. 53-pg., 2025-06-01.

Resumo

Peixoto's stability theorem stands as a cornerstone in the global dynamical examination of flows on smooth two-manifolds, a significant landmark in Dynamical Systems research. This theorem has served as a blueprint for subsequent global classification theorems within the field. Building upon Peixoto's foundational work, Gutierrez and Sotomayor introduced a compelling generalization and their contribution extends Peixoto's conditions for structural stability of C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>{1}$$\end{document}-vector fields on smooth surfaces to encompass singular two-manifolds M. Furthermore, generalizing this classical theorem to varied and richer topological configurations such as these non-smooth surfaces which feature singular loci comprising cones (C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}), cross-caps (W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{W}}$$\end{document}), double (D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document}), and triple points (T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{T}}$$\end{document}) marks a milestone for research in Singular Dynamics. In homage to their contributions, we have named this class of dynamical systems as Gutierrez-Sotomayor flows, GS flows for short. It is our intent, in this article to produce a survey of the state of the art for GS flows which have garnered significant attention in the past years. Our interest is two-fold: firstly present a local and global analysis of GS flows phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document} on singular surfaces M and secondly describe the effects of homotopical deformations on (phi,M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varphi , M)$$\end{document} which are in correspondence to a spectral sequence of an associated chain complex for phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}. Herein we address the far reaching results that are obtained by using Spectral Sequence Theory which has yielded several homotopical cancellation theorems within the dynamics. (AU)

• Processo FAPESP: 23/03430-8 - Condições suficientes para a realização de grafos de Lyapunov como fluxos Gutierrez-Sotomayor
• Beneficiário: Murilo André de Jesus Zigart
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado
• Processo FAPESP: 22/16455-6 - Topologia algébrica, geométrica e diferencial
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 23/17645-6 - Teoria de Morse-Conley, Variedades Singulares e Homologia de Intersecção
• Beneficiário: Dahisy Valadão de Souza Lima
• Modalidade de apoio: Bolsas no Exterior - Pesquisa
• Processo FAPESP: 18/13481-0 - Geometria de sistemas de controle, sistemas dinâmicos e estocásticos
• Beneficiário: Marco Antônio Teixeira
• Modalidade de apoio: Auxílio à Pesquisa - Temático