An algebraic-topological approach to dynamical systems and symplectic topology
Qualitative theory of differential equations and singularity theory
Morse-Conley Theory, Singular Varieties, and Intersection Homology
| Full text | |
| Author(s): |
Total Authors: 3
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| Affiliation: | [1] Univ Fed ABC, CMCC, Santo Andre, SP - Brazil
[2] Univ Estadual Campinas, IMECC, Campinas, SP - Brazil
Total Affiliations: 2
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| Document type: | Journal article |
| Source: | JOURNAL OF SINGULARITIES; v. 23, p. 33-91, 2021. |
| Web of Science Citations: | 0 |
| Abstract | |
In this article, we present a dynamical homotopical cancellation theory for Gutierrez-Sotomayor singular flows phi, GS-flows, on singular surfaces M. This theory generalizes the classical theory of Morse complexes of smooth dynamical systems together with the corresponding cancellation theory for non-degenerate singularities. This is accomplished by defining a GS-chain complex for (M, phi) and computing its spectral sequence (E-r, d(r)). As r increases, algebraic cancellations occur, causing modules in E-r to become trivial. The main theorems herein relate these algebraic cancellations within the spectral sequence to a family [M-r, phi(r)] of GS-flows phi(r) on singular surfaces M-r, all of which have the same homotopy type as M. The surprising element in these results is that the dynamical homotopical cancellation of GS-singularities of the flows phi(r) are in consonance with the algebraic cancellation of the modules in E-r of its associated spectral sequence. Also, the convergence of the spectral sequence corresponds to a GS-flow phi((r) over bar) on M-(r) over bar, for some (r) over bar, with the property that phi((r) over bar) admits no further dynamical homotopical cancellation of GS-singularities. (AU) | |
| FAPESP's process: | 14/11943-6 - Spectral sequences for Morse-Bott and Morse-Novikov flows study |
| Grantee: | Dahisy Valadão de Souza Lima |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 18/13481-0 - Geometry of control, dynamical and stochastic systems |
| Grantee: | Marco Antônio Teixeira |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 15/10930-0 - Symplectic dynamics and spectral sequences |
| Grantee: | Dahisy Valadão de Souza Lima |
| Support Opportunities: | Scholarships abroad - Research Internship - Post-doctor |
| FAPESP's process: | 16/24707-4 - Algebraic, geometric and differential topology |
| Grantee: | Daciberg Lima Gonçalves |
| Support Opportunities: | Research Projects - Thematic Grants |