Plane foliations with a saddle singularity

Gutiérrez, Carlos; Martínez-Alfaro, José; Venato-Santos, Jean

doi:10.1016/j.topol.2011.09.023

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Cited by 2 publications

(3 citation statements)

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“…Indeed, in [18], Olech had already proved the equivalence between Markus-Yamabe Conjecture and the injectivity of F in R 2 . Recently, in [12] and [22], analogous results were obtained when the equilibrium point 0 is not an attractor, but a hyperbolic saddle and a center, respectively.…”

Section: Introductionmentioning

confidence: 57%

Half-Reeb components, Palais-Smale condition and global injectivity of local diffeomorphisms in $\mathbb{R}^3$

Braun

Venato-Santos

2014

Publicacions Matemàtiques

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Let F = (F 1 , F 2 , F 3 ) : R 3 → R 3 be a C ∞ local diffeomorphism. We prove that each of the following conditions are sufficient to the global injectivity of F :A) The foliations F F i made up by the connected components of the level surfaces F i = constant, consist of leaves without half-Reeb components induced byWe also prove that B) implies A) and give examples to show that the converse is not true. Further, we give examples showing that none of these conditions is necessary to the global injectivity of F .

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Section: Introductionmentioning

confidence: 57%

Half-Reeb components, Palais-Smale condition and global injectivity of local diffeomorphisms in $\mathbb{R}^3$

Braun

Venato-Santos

2014

Publicacions Matemàtiques

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“…An interesting discussion, as well as illustrations, for the continuous-time case may be found in [13].…”

Section: Background and Definitionsmentioning

confidence: 99%

“…Interest in this has then extended to planar discrete dynamics. The case when the unique fixed point is a local saddle was addressed in [13] for C 1 vector fields. However, the problem of characterising the existence of a global saddle for planar diffeomorphisms is still open.…”

Section: Introductionmentioning

confidence: 99%