Continuous flows generate few homeomorphisms

Bonomo, Wescley; Varandas, Paulo

doi:10.1017/s0013091520000280

Cited by 5 publications

(1 citation statement)

References 31 publications

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“…A candidate will be the time-one map because it corresponds to the discretization functor, which has been considered in [5]. However, it is known that this construction is not very expressive, and it is unclear whether an interesting equivalence can be found [3].…”

Section: Discussionmentioning

confidence: 99%

Poincaré maps and suspension flows: a categorical remark

Suda

2021

Preprint

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Poincaré maps and suspension flows are examples of fundamental constructions in the study of dynamical systems. This study aimed to show that these constructions define an adjoint pair of functors if categories of dynamical systems are suitably set. First, we consider the construction of Poincaré maps in the category of flows on topological manifolds, which are not necessarily smooth. We show that well-known results can be generalized and the construction of Poincaré maps is functorial, if a category of flows with global Poincaré sections is adequately defined. Next, we consider the construction of suspension flows and its functoriality. Finally, we consider the adjointness of the constructions of Poincaré maps and suspension flows. By considering the naturality, we can conclude that the concepts of topological equivalence or topological conjugacy of flows are not sufficient to describe the correspondence between map dynamical systems and flows with global Poincaré sections. We define another category of flows with global Poincaré sections and show that the suspension functor and the Poincaré map functor form an adjoint equivalence if these categories are considered. Hence, a categorical correspondence between map dynamical systems and flows with global Poincaré sections is obtained. This will enable us to better understand the connection between map dynamical systems and flows.

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Section: Discussionmentioning

confidence: 99%

Poincaré maps and suspension flows: a categorical remark

Suda

2021

Preprint

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Symmetries of $$C^r$$-vector Fields on Surfaces

Bonomo,

Rocha,

Varandas

2023

Bull Braz Math Soc, New Series

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Shape and stability of rotation sets for incompressible flows on the torus

Bonomo,

Lima,

Varandas

2024

Isr. J. Math.

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In this paper we introduce the concept of rotation cones, determined by rotation sets, and use these to describe the stability of rotation sets of incompressible and fixed-point free continuous flows on the torus $$\mathbb{T}^{d}$$ T d , d ⩾ 2. The previous concept is equivalent and extends the stability of rotation numbers (and rotation vectors) in the special case of fixed-point free continuous flows on $$\mathbb{T}^{2}$$ T 2 . We prove that incompressible Lipschitz vector fields with stable rotation sets have convex rotation cones with non-empty interior, and that all extremal edges are collinear with vectors in ℤd. If, in addition, the rotation cones are proper, then these are polyhedral with finitely many edges. Finally we prove that the set of vector fields with stable rotation sets is C0-open and dense among those having proper rotation cones.

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Continuous flows generate few homeomorphisms

Cited by 5 publications

References 31 publications

Poincaré maps and suspension flows: a categorical remark

Poincaré maps and suspension flows: a categorical remark

Symmetries of $$C^r$$-vector Fields on Surfaces

Shape and stability of rotation sets for incompressible flows on the torus

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