Kinematic expansive flows

Artigue, Alfonso

doi:10.1017/etds.2014.65

Cited by 22 publications

(15 citation statements)

References 17 publications

(50 reference statements)

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“…Definition 3.4. [2] We say that (φ t ) t∈R is kinematic expansive if for each ε > 0, there exists δ > 0 with the following property. If…”

Section: Concepts and Main Resultsmentioning

confidence: 99%

“…We consider the horocycle flow on compact Riemann surfaces of constant negative curvature. It is well known that any compact orientable surface with constant negative curvature is isometric to a factor \H 2 = { z, z ∈ H 2 }, where is a discrete subgroup of the group PSL(2, R) = SL(2, R)/{±E 2 }; here SL(2, R) is the group of all real 2 × 2 matrices with unit determinant, E 2 is the unit matrix and H 2 denotes the hyperbolic plane which is the upper half plane {(x, y) ∈ R 2 : y > 0}, endowed with the hyperbolic metric…”

Section: Compact Riemann Surfaces Of Constant Negative Curvaturementioning

confidence: 99%

“…Expansive flows have been studied since 1972 by several mathematicians with different varieties of expansiveness [1,2,4,6,10]. Bowen and Walters [4] and Flinn [6] introduced the term expansive flow with different definitions which are equivalent to each other in the case of fixed-point-free flows.…”

Section: Introductionmentioning

confidence: 99%

“…Such a flow is called positive strong separating. In 2014, Artigue [2] provided the concept of kinematic expansiveness which implies separation. The author also gave the notions of strong kinematic expansiveness, geometric separation, as well as examples to analyze relationships among the concepts.…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, a flow is strong kinematic expansive if all its time changes are kinematic expansive. Note that the terms strong kinematic expansiveness in the sense of Artigue [2] and weak expansiveness in the sense of Flinn [6] are equivalent. In a recent paper, Huynh [8] shows that the horocycle flow on compact Riemann surfaces of constant negative curvature is kinematic expansive but not geometric separating (and hence it is not expansive in the sense of Bowen and Walters).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A solution to Flinn’s conjecture on weakly expansive flows

Hien¹

2020

Ergod. Th. Dynam. Sys.

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“…Definition 3.4. [2] We say that (φ t ) t∈R is kinematic expansive if for each ε > 0, there exists δ > 0 with the following property. If…”

Section: Concepts and Main Resultsmentioning

confidence: 99%

Section: Compact Riemann Surfaces Of Constant Negative Curvaturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A solution to Flinn’s conjecture on weakly expansive flows

Hien¹

2020

Ergod. Th. Dynam. Sys.

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The centralizer of Komuro-expansive flows and expansive $${{\mathbb {R}}}^d$$ R d actions

2017

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In this paper we study the centralizer of flows and R d -actions on compact Riemannian manifolds. We prove that the centralizer of every C ∞ Komuro-expansive flow with non-resonant singularities is trivial, meaning it is the smallest possible, and deduce there exists an open and dense subset of geometric Lorenz attractors with trivial centralizer. We show that R d -actions obtained as suspension of Z d -actions are expansive if and only if the same holds for the Z d -actions. We also show that homogeneous expansive R d -actions have quasi-trivial centralizers, meaning that it consists of orbit invariant, continuous linear reparameterizations of the R d -action. In particular, homogeneous Anosov R d -actions have quasi-trivial centralizer. Expansive flows and trivial centralizers and R d -actions and singular-hyperbolicity and geometric Lorenz attractors 1

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On the centralizer of vector fields: criteria of triviality and genericity results

Obata

Santiago

2020

Math. Z.

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In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold M d has a "small" centralizer. In the C 1 case, we give two criteria, one of which is C 1 -generic, which guarantees that the centralizer of a C 1 -generic vector field is indeed small, namely collinear. The other criterion states that a C 1 separating flow has a collinear C 1 -centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as quasi-triviality. In particular, the C 1 -centralizer of a C 1 -generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a C 1 -generic vector field, which includes C 1 -generic Axiom A (or sectional Axiom A) vector fields and C 1 -generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is trivial (as small as it can be), and we show that in higher regularity, collinearity and triviality of the C d -centralizer are equivalent properties for a generic vector field in the C d topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity C 2 , the C 1 -centralizer is trivial. 9 4. Quasi-triviality 13 4.1. Collinear does not imply quasi-trivial 13 4.2. The case of hyperbolic zeros 13 5. The study of invariant functions and trivial centralizers 18 5.1. First integrals and trivial C 1 -centralizers 19 5.2. Some results in higher regularity 20 6. The generic case 23 Appendix A. The separating property is not generic 40 References

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Kinematic expansive flows

Cited by 22 publications

References 17 publications

A solution to Flinn’s conjecture on weakly expansive flows

A solution to Flinn’s conjecture on weakly expansive flows

The centralizer of Komuro-expansive flows and expansive $${{\mathbb {R}}}^d$$ R d actions

On the centralizer of vector fields: criteria of triviality and genericity results

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