Entropy of the geodesic flow for metric spaces and Bruhat–Tits buildings

Leuzinger, Enrico

doi:10.1515/advgeom.2006.029

Cited by 9 publications

(8 citation statements)

References 19 publications

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“…Corollary 1.2 is a version of Manning's [13] analogous result for Riemannian manifolds of non-positive curvature, and relies on some work of Leuzinger [12]. This is discussed in the nal section of the paper, where we also establish the following entropy rigidity result for the geodesic ow.…”

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confidence: 83%

“…Theorem 2 is a version of Manning's [Man79] analogous result for Riemannian manifolds of non-positive curvature, and relies on some work of Leuzinger [Leu06]. Theorem 3 follows from our Theorem 2 and a rigidity result of Bourdon ([Bou96]).…”

Section: Introductionmentioning

confidence: 88%

“…Theorem 21 (Leuzinger,[Leu06]; compare with Manning [Man79]). Let ( X, d) be a geodesically complete, locally geodesic metric space endowed with a measure vol, and having compact quotient X Γ.…”

Section: Local Geometry Of Cat(−1) Spaces In Higher Dimensionsmentioning

confidence: 99%

“…Theorem 1.3 is a reformulation of a rigidity result of Bourdon ([2]). Our main observation is to note how, using Leuzinger's work [12], Bourdon's theorem can be recast as a topological entropy rigidity theorem. This fact may well be known to experts, but we have not found it addressed in the literature.…”

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confidence: 99%

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On the Hausdorff Dimension of CAT(κ) Surfaces

Constantine

Lafont

2016

Analysis and Geometry in Metric Spaces

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Abstract:We prove that a closed surface with a CAT(κ) metric has Hausdor dimension = , and that there are uniform upper and lower bounds on the two-dimensional Hausdor measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic ow for such surfaces. Finally, we give a short proof of topological entropy rigidity for geodesic ow on certain CAT(− ) manifolds.

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confidence: 83%

Section: Introductionmentioning

confidence: 88%

Section: Local Geometry Of Cat(−1) Spaces In Higher Dimensionsmentioning

confidence: 99%

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confidence: 99%

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On the Hausdorff Dimension of CAT(κ) Surfaces

Constantine

Lafont

2016

Analysis and Geometry in Metric Spaces

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“…Finally, the volume entropy of hyperbolic buildings considers the asymptotic growth of volumes of balls in the building (by analogy with volume entropy for Riemannian manifolds). This topic has been investigated by Hersonsky-Paulin [37], Leuzinger [45] and most thoroughly by Ledrappier-Lim [44], using the geodesic flow on apartments and measures on suitable boundaries.…”

Section: Property (T) and Finiteness Properties Ballmann-świ ֒mentioning

confidence: 99%

Lattices in hyperbolic buildings

Thomas¹

2016

Geometry, Topology, and Dynamics in Negative Curvature

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1.1. Coxeter groups and Coxeter polytopes. We mostly follow the reference Davis [19], particularly Chapter 6, and concentrate on the hyperbolic case.Recall that a Coxeter group is a group W with finite generating set S and presentation of the form W = s ∈ S | (st) mst = 1 where s, t ∈ S, m ss = 1 for all s ∈ S and if s = t then m st is an integer ≥ 2 or m st = ∞, meaning that the product st has infinite order. The pair (W, S) is called a Coxeter system. A Coxeter system (W, S) is right-angled if for each s, t ∈ S with s = t, m st ∈ {2, ∞}. Note that m st = 2 if and only if st = ts. Let X n be the n-dimensional sphere, n-dimensional Euclidean space or n-dimensional (real) hyperbolic space. Many important examples of Coxeter groups arise as discrete reflection groups acting on X n , as follows. Let P be a convex polyhedron in X n with all dihedral angles integer submultiples of π. Such a P is called a Coxeter polytope. Let W = W (P ) be the group generated by the set S = S(P ) of reflections in the codimension one faces of P . Then (W, S) is a Coxeter system and W is a discrete subgroup of the isometry group of X n (see [19, Theorem 6.4.3]). Moreover, the action of W tessellates X n by copies of P . For example, let P be a right-angled hyperbolic p-gon, p ≥ 5. Then the corresponding Coxeter system is right-angled with p generators, one for each side of P , so that if s and t are reflections in distinct sides then m st = 2 when these sides are adjacent, and otherwise m st = ∞.Suppose that X n is the sphere or Euclidean space. Then Coxeter polytopes P ⊂ X n exist and have been classified in every dimension, and the corresponding Coxeter systems (W, S) are the spherical or affine Coxeter systems, respectively (see [19, Table 6.1] for the classification).If X n is n-dimensional hyperbolic space H n , then there is no complete classification of Coxeter polytopes. Vinberg's Theorem [65] establishes that compact hyperbolic Coxeter polytopes can exist only in dimension n ≤ 29, although at the time of writing the highest dimension in which an example is known is n = 8 (due to Bugaenko [12]). Finite volume hyperbolic Coxeter polytopes have also been investigated, with for example Prokhorov [50] proving these can exist only in dimension n ≤ 995. For n ≤ 6 (respectively, n ≤ 19), there are infinitely many essentially distinct compact (respectively, finite volume) hyperbolic Coxeter polytopes (Allcock [2]). In dimension 3, Andreev's Theorem [3] classifies compact hyperbolic Coxeter polytopes, but in dimensions n ≥ 4 only special cases have been considered, and there seems little hope of a complete list. Hyperbolic Coxeter polytopes which are simplices exist in dimensions n ≤ 4 only, and their classification is given in [19, Table 6.2]. Right-angled compact hyperbolic polytopes also exist in dimensions n ≤ 4 only, and there are infinitely many examples in each dimension n ≤ 4 (see [66]). A right-angled example in dimension 3 is the dodecahedron, which tessellates H 3 as depicted on the cover of Thurston's book [59], and a right-...

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