Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology

Cuesta, Fernando Alcalde; Dal’Bo, Françoise; Martínez, Miguel; Verjovsky, Alberto

doi:10.3934/dcds.2016001

Cited by 14 publications

(50 citation statements)

References 23 publications

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“…A companion paper [2] by the first two authors will address the case where the generic leaf has a Cantor set of ends. It contains a refinment of [1,Theorem 2]: for every leaf of such a lamination, all isolated ends are accumulated by genus -this is called condition ( * ) in [2]. Using the formalism developped in the present paper, as well as new techniques, it is proven that this is the only obstruction.…”

Section: Introductionmentioning

confidence: 90%

“…It is worth mentioning the Ghys-Kenyon example, constructed in [24] which is a minimal lamination containing leaves with different conformal types. Also one has the famous Hirsch's foliation where all leaves have infinite topological types (see [26] for the original construction and [1,3,13,23] for the minimal construction). Other aspects of the study of this subject have been pursued in different works, a non exhaustive list is [16,17,21,31,35,37,38].…”

Section: Introductionmentioning

confidence: 99%

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Topology of leaves for minimal laminations by hyperbolic surfaces

Alvarez¹,

Brum²,

Martínez³

et al. 2019

Preprint

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We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the second systole of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Topology of leaves for minimal laminations by hyperbolic surfaces

Alvarez¹,

Brum²,

Martínez³

et al. 2019

Preprint

Self Cite

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“…In general, the minimality of the foliation F does not suffice to obtain the minimality of the horocycle flow h + s as in Hedlund's theorem (see examples in [3] and [19]). This question has been addressed in [3], [4] and [22], obtaining several results under certain restrictions. These are described more concretely below.…”

Section: Introductionmentioning

confidence: 99%

“…by the trivial fibration D : H × G → G. (2) In [4], the minimality of the horocycle flow is proved for minimal foliations that have a loop which is non-homotopic to zero in its leaf and which has trivial holonomy.…”

Section: Introductionmentioning

confidence: 99%

Unique ergodicity of the horocycle flow on Riemannnian foliations

Cuesta¹,

Dal’Bo²,

Martínez³

et al. 2018

Ergod. Th. Dynam. Sys.

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A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.

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“…Finally, we ask the following question: does the minimality of the B -action imply the minimality of the horocycle flow? There is strong evidence that the answer might be positive, see [1], [2], and [19].…”

Section: Introductionmentioning

confidence: 99%

Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

Martínez¹,

Matsumoto²,

Verjovsky³

2016

JMD

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We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle (M , T 1 F ) of a compact minimal lamination (M , F ) by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if F has a leaf which is not simply connected, the horocyle flow is topologically transitive.

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Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology

Cited by 14 publications

References 23 publications

Topology of leaves for minimal laminations by hyperbolic surfaces

Topology of leaves for minimal laminations by hyperbolic surfaces

Unique ergodicity of the horocycle flow on Riemannnian foliations

Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

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