Sur l’inexistence d’ensembles minimaux pour le flot horocyclique

Gaye, Masseye; Lo, C M M

doi:10.5802/cml.37

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“…The first example of such a surface was given by M. Kulikov in [Kul04]. Later, theorems of non-existence were obtained in [Mat16] and [GL17].…”

Section: Introductionmentioning

confidence: 99%

On the links between horocyclic and geodesic orbits on geometrically infinite surfaces

Bellis¹

2018

Journal De L’École Polytechnique — Mathématiques

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We study the topological dynamics of the horocycle flow h R on a geometrically infinite hyperbolic surface S. Let u be a non-periodic vector for h R in T 1 S. Suppose that the half-geodesic u(R + ) is almost minimizing and that the injectivity radius along u(R + ) has a finite inferior limit Inj(u(R + )). We prove that the closure of h R u meets the geodesic orbit along un unbounded sequence of points gt n u. Moreover, if Inj(u(R + )) = 0, the whole half-orbit g R + u is contained in h R u.When Inj(u(R + )) > 0, it is known that in general g R + u ⊂ h R u. Yet, we give a construction where Inj(u(R + )) > 0 and g R + u ⊂ h R u, which also constitutes a counter-example to Proposition 3 of [Led97].

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“…The first example of such a surface was given by M. Kulikov in [Kul04]. Later, theorems of non-existence were obtained in [Mat16] and [GL17].…”

Section: Introductionmentioning

confidence: 99%