RECORD
Star vector fields on three-manifolds are multi-singular hyperbolic
Dynamical Systems · Submitted 14 Apr 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this article, we prove that a vector field exhibiting all periodic orbits robustly of the same index is multi-singular hyperbolic, generalizing the previous results.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits robustly hyperbolic (star flows), are hyperbolic in absence of singularities. Morales, Pacifico and Pujals proposed a partial hyperbolicity called "singular hyperbolicity" that characterizes an open and dense subset of three dimensional star flows. In higher dimensions, Bonatti and da Luz characterize an open and dense set of star vector fields by multi-singular hyperbolicity. In this article, we prove that a vector field exhibiting all periodic orbits robustly of the same index is multi-singular hyperbolic, generalizing the previous results. As a corollary, we obtained that all three-dimensional star flows are multi-singular hyperbolic. Moreover, if all singularities in the same class exhibit the same index, the star flow is singular hyperbolic. Additionally, star flows with robust chain recurrence classes in any dimension are multisingular hyperbolic.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. | background 0.89 | 0.04 | 0.01 | asserted 1.00 |
| s1 | Notably, the vector fields with all periodic orbits robustly hyperbolic (star flows), are hyperbolic in absence of singularities. | background 1.00 | 0.07 | 0.03 | asserted 1.00 |
| s2 | Morales, Pacifico and Pujals proposed a partial hyperbolicity called "singular hyperbolicity" that characterizes an open and dense subset of three dimensional star flows. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s3 | In higher dimensions, Bonatti and da Luz characterize an open and dense set of star vector fields by multi-singular hyperbolicity. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s4 | In this article, we prove that a vector field exhibiting all periodic orbits robustly of the same index is multi-singular hyperbolic, generalizing the previous results. | result 0.61 | 0.95 | 0.90 | asserted 1.00 |
| s5 | As a corollary, we obtained that all three-dimensional star flows are multi-singular hyperbolic. | result 0.97 | 0.93 | 0.36 | asserted 1.00 |
| s6 | Moreover, if all singularities in the same class exhibit the same index, the star flow is singular hyperbolic. | result 0.99 | 0.78 | 0.06 | asserted 1.00 |
| s7 | Additionally, star flows with robust chain recurrence classes in any dimension are multisingular hyperbolic. | result 0.99 | 0.76 | 0.10 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits robustly hyperbolic (star flows), are hyperbolic in absence of singularities. Morales, Pacifico and Pujals proposed a partial hyperbolicity called "singular hyperbolicity" that characterizes an open and dense subset of three dimensional star flows. In higher dimensions, Bonatti and da Luz characterize an open and dense set of star vector fields by multi-singular hyperbolicity. In this article, we prove that a vector field exhibiting all periodic orbits robustly of the same index is multi-singular hyperbolic, generalizing the previous results. As a corollary, we obtained that all three-dimensional star flows are multi-singular hyperbolic. Moreover, if all singularities in the same class exhibit the same index, the star flow is singular hyperbolic. Additionally, star flows with robust chain recurrence classes in any dimension are multisingular hyperbolic.
Human review
References
How the literature uses it
1 arXiv paper cite this in 1 sentence
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.