Recurrent geodesics and controlled concentration points

Aebischer, Beat; Hong, Sungbok; McCullough, Darryl

doi:10.1215/s0012-7094-94-07523-6

Cited by 5 publications

(18 citation statements)

References 8 publications

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“…The latter property for ξ is equivalent to the existence of a recurrent geodesic γ with γ (+∞) = ξ and γ (−∞) = η. Hence we obtain the following connection between recurrent geodesics and controlled concentration points which also holds for manifolds (see [1]). …”

Section: Proof Of Main Theoremmentioning

confidence: 65%

“…Denote by ∆ = δ (0), δ (l ), δ (l + ε) the corresponding comparison triangle which is non-degenerate by inequality (1). Choose points B on δ | [0,l] and B on δ | [l,l+ε] such that d (B, δ (l )) = d (B , δ (l )) = ε < ε and denote by B and B the corresponding points on the comparison triangle.…”

Section: ])mentioning

confidence: 99%

“…It is shown in this work that, partially, this is the case with the notion of recurrence. For complete hyperbolic manifolds, a recent result of Aebischer, Hong and McCullough (see [1]) states that a geodesic is recurrent if and only if it is approximated by closed geodesics. We show that, in metric spaces with curvature less than or equal to χ, χ < 0, recurrent geodesics are approximated by closed geodesics (see Theorem 2 below).…”

Section: Introductionmentioning

confidence: 99%

“…The notion of convergence in the above definition is analogous to the tangential condition which defines recurrence in the manifold case. We use the notion of approximation given in Definition 6 below which was introduced in [1] in order to characterize recurrent geodesics in hyperbolic manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Approximation of recurrence in negatively curved metric spaces

Charitos¹,

Tsapogas²

2000

Pacific J. Math.

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Section: Proof Of Main Theoremmentioning

confidence: 65%

Section: ])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximation of recurrence in negatively curved metric spaces

Charitos¹,

Tsapogas²

2000

Pacific J. Math.

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“…For example, one of several such characterizations is that there exist points q = r in S m−1 and a sequence of distinct elements γ n ∈ Γ such that γ n (p) → q and γ n (x) → r for every x ∈ S m−1 − {p}. For other topological characterizations of conical limit points, see [1,3,5].…”

Section: Introductionmentioning

confidence: 99%

Concentration points for Fuchsian groups

Hong

McCullough

2000

Topology and its Applications

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A limit point p of a discrete group of Möbius transformations acting on S n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S n at p. For the case of Fuchsian groups (n = 1), every concentration point is a conical limit point, but even for finitely generated groups not every conical limit point is a concentration point. A slightly weaker concentration condition is given which is satisfied if and only if p is a conical limit point, for finitely generated Fuchsian groups. In the infinitely generated case, it implies that p is a conical limit point, but not all conical limit points satisfy it. Examples are given that clarify the relations between various concentration conditions.Definition: An open set U in S m−1 can be concentrated at p if for every neighborhood V of p, there exists an element γ ∈ Γ such that p ∈ γ(U) and γ(U) ⊆ V . If in addition the element γ can always be selected so that p ∈ γ(V ), then one says that U can be concentrated with control.Note that U can be concentrated at p if and only if the set of translates of U contains a local basis for the topology of S m−1 at p. Also, one can easily check from the definition that (1) there exists a neighborhood of p which can be concentrated with control if and only if there is a connected neighborhood which can be concentrated with control (take the connected component of U that contains p, and require that γ −1 (p) ∈ U ∩ V ), and (2) if a neighborhood of p can be concentrated with control, then every smaller neighborhood can be concentrated with control.Definition: The limit point p is called a controlled concentration point for §1. Introduction 3 Γ if it has a neighborhood which can be concentrated with control at p.Concentration with control is studied in [1]. Analogously to conical limit points, p is a controlled concentration point if and only if there exist a point r = p in S m−1 and a sequence γ n of distinct elements of Γ so that γ n (p) → p and γ n (x) → r for all x ∈ S m−1 − {p}. In particular, every controlled concentration point is a conical limit point. However, examples are given in [1] of conical limit points of 2-generator Schottky groups which are not controlled concentration points (see also proposition 4.1 below). For groups of divergence type, controlled concentration points have full Patterson-Sullivan measure in the limit set. There is a direct connection between controlled concentration points and the dynamics of geodesics in the hyperbolic manifold B m /Γ. Call a geodesic ray in B m /Γ recurrent if it is the image of a geodesic ray in B m that ends at a controlled concentration point. In an appropriate metric, the space of recurrent geodesic rays in B m /Γ is a metric completion of the space of closed geodesics in B m /Γ (where both spaces are topologized as subspaces of the unit tangent bundle of B m /Γ).We turn now to weaker concentration properties. It is not difficult to show (see [6]) that every limit point p ...

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Conical limit points and the Cannon-Thurston map

Jeon

Kapovich

Leininger

et al. 2016

Conform. Geom. Dyn.

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Abstract. Let G be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space Z so that there exists a continuous G-equivariant map i : ∂G → Z, which we call a Cannon-Thurston map. We obtain two characterzations (a dynamical one and a geometric one) of conical limit points in Z in terms of their pre-images under the Cannon-Thurston map i. As an application we prove, under the extra assumption that the action of G on Z has no accidental parabolics, that if the map i is not injective then there exists a non-conical limit point z ∈ Z with |i −1 (z)| = 1. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of wordhyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if G is a non-elementary torsion-free word-hyperbolic group then there exists x ∈ ∂G such that x is not a "controlled concentration point" for the action of G on ∂G.

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Recurrent geodesics and controlled concentration points

Cited by 5 publications

References 8 publications

Approximation of recurrence in negatively curved metric spaces

Approximation of recurrence in negatively curved metric spaces

Concentration points for Fuchsian groups

Conical limit points and the Cannon-Thurston map

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