Geodesic Flow on Ideal Polyhedra

Charitos, Charalampos; Tsapogas, Georgios

doi:10.4153/cjm-1997-033-8

Cited by 5 publications

(17 citation statements)

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“…In this section we will establish this property without dealing with its equivalence to topological transitivity. For the proof of the above proposition we will need the following result: The proof of this proposition is given in [10] for ideal polyhedra but it applies verbatim to our context. Now let f, g ∈ GY be arbitrary.…”

Section: Topological Transitivitymentioning

confidence: 99%

“…Certain properties of this class of spaces, including transitivity of the geodesic flow, have been studied in [9], [10], [11] and [12].…”

Section: Theorem 54 Let Y Be a Negatively Curved Polyhedron Which Imentioning

confidence: 99%

“…10 in [10]) and π 1 (Y ) is a non-elementary group of isometries acting properly discontinuously on Y (see Cor. 12 in [10]). Moreover, proposition 5.5 applies verbatim to ideal polyhedra.…”

Section: Theorem 54 Let Y Be a Negatively Curved Polyhedron Which Imentioning

confidence: 99%

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Topological mixing in 𝐶𝐴𝑇(-1)-spaces

Charitos

Tsapogas

2001

Trans. Amer. Math. Soc.

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Abstract. If X is a proper CAT (−1)-space and Γ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y = X/Γ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary ∂X and the non-wandering set of the flow equals the whole quotient space of geodesics GY := GX/ Γ (the latter being redundant when Y is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete CAT (−1)-spaces by a one-ended group of isometries and (C) finite n-dimensional ideal polyhedra.

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Section: Topological Transitivitymentioning

confidence: 99%

“…Certain properties of this class of spaces, including transitivity of the geodesic flow, have been studied in [9], [10], [11] and [12].…”

Section: Theorem 54 Let Y Be a Negatively Curved Polyhedron Which Imentioning

confidence: 99%

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Topological mixing in 𝐶𝐴𝑇(-1)-spaces

Charitos

Tsapogas

2001

Trans. Amer. Math. Soc.

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“…The two-dimensional simplicial complexes defined above can be considered as a generalization of ideal polyhedra of dimension 2 which are built by gluing hyperbolic ideal triangles, see [3,5,6].…”

Section: Definitionmentioning

confidence: 99%

“…In this section, by using the methods of [5], we will prove that the limit set of the action of G on X is equal to ∂ X.…”

Section: The Limit Set Of π 1 (X)mentioning

confidence: 99%

On the boundary of a special class of hyperbolic two-dimensional simplicial complexes

2009

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A class of hyperbolic two-dimensional complexes X is defined. It is shown that the limit set of the action of π1(X) on the universal covering X of X is equal to the visual boundary ∂ X of and that ∂ X is path connected and locally path connected. A kind of Sierpinski set is described which is homeomorphic to the visual boundary of certain ideal polyhedra. Mathematics Subject Classification (2000). 57M20, 53C23.

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