On the links between horocyclic and geodesic orbits on geometrically infinite surfaces
[Sur les liens entre les orbites horocycliques et géodésiques sur les surfaces géométriquement infinies]
Bellis, Alexandre1
1 Institut de Recherche Mathématique de Rennes, Université de Rennes 1 263 Avenue du Général Leclerc, 35042 Rennes, France
Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 443-454.

Nous étudions l’intersection entre une demi-géodésique quasi-minimisante et l’adhérence de l’orbite horocyclique correspondante sur une surface hyperbolique lisse géométriquement infinie. Nous démontrons que si la demi-géodésique traverse un nombre infini de parties de la surface de rayons d’injectivité bornés supérieurement, alors l’intersection contient une suite non bornée d’éléments. Nous démontrons aussi que si la demi-géodésique traverse des parties arbitrairement fines de la surface, l’intersection est toute la demi-géodésique. Enfin, nous construisons un exemple montrant que cette dernière condition n’est pas nécessaire.

We study the intersection between an almost minimizing half-geodesic and the closure of the corresponding horocyclic orbit on a smooth geometrically infinite surface. We prove that if the half-geodesic goes through an infinite number of parts of the surface with injectivity radii bounded from above, then the intersection contains an unbounded sequence of points. We also prove that if the half-geodesic goes through arbitrarily thin parts of the surface, the intersection is the whole half-geodesic. Finally, we construct an example proving that this last condition is not necessary.

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DOI : 10.5802/jep.75
Classification : 37D40, 37E35
Keywords: Topological dynamics, flows on surfaces
Mot clés : Dynamique topologique, flots sur des surfaces
Bellis, Alexandre 1

1 Institut de Recherche Mathématique de Rennes, Université de Rennes 1 263 Avenue du Général Leclerc, 35042 Rennes, France 
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Bellis, Alexandre. On the links between horocyclic and geodesic orbits on geometrically infinite surfaces. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 443-454. doi : 10.5802/jep.75. http://www.numdam.org/articles/10.5802/jep.75/

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