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The PattersonSullivan embedding and minimal volume entropy for Outer space
 Geom. Funct. Anal. (GAFA
"... Abstract. Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the PattersonSullivanBowenMargulis map from the CullerVogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding and ..."
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Cited by 21 (14 self)
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Abstract. Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the PattersonSullivanBowenMargulis map from the CullerVogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding
c © Birkhäuser Verlag, Basel 2007 GAFA Geometric And Functional Analysis THE PATTERSON–SULLIVAN EMBEDDING AND MINIMAL VOLUME ENTROPY FOR OUTER SPACE
"... Abstract. Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding an ..."
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Abstract. Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding
The actions of Out(Fk) on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility
, 2006
"... We prove that for k ≥ 5 there does not exist a continuous map ∂CV (Fk) → PCurr(Fk) that is either Out(Fk)equivariant or Out(Fk)antiequivariant. Here ∂CV (Fk) is the “lengthfunction” boundary of CullerVogtmann’s Outer space CV (Fk), and PCurr(Fk) is the space of projectivized geodesic currents ..."
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Cited by 25 (14 self)
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of CV (Fk) into PCurr(Fk) (such as the PattersonSullivan embedding) produces a new compactification of Outer space, different from the usual “lengthfunction” compactification CV (Fk) = CV (Fk) ∪ ∂CV (Fk).
4. Busemann Functions for the Teichmuller Metric 7
"... Abstract. For a convex cocompact subgroup G < Mod(S), and points x, y ∈ Teich(S) we obtain asymptotic formulas as R→ ∞ of BR(x)∩Gy  as well as the number of conjugacy classes of pseudoAnosov elements in G of dilatation at most R. We do this by developing an analogue of PattersonSullivan theor ..."
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Abstract. For a convex cocompact subgroup G < Mod(S), and points x, y ∈ Teich(S) we obtain asymptotic formulas as R→ ∞ of BR(x)∩Gy  as well as the number of conjugacy classes of pseudoAnosov elements in G of dilatation at most R. We do this by developing an analogue of PattersonSullivan
Zeta functions that hear the shape of a Riemann surface
, 2007
"... Gunther Cornelissen and Matilde Marcolli To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose “Riemannian ” aspect (Hilbert space and Dirac operator) encode the bound ..."
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the boundary action through its PattersonSullivan measure. We prove that the ergodic rigidity theorem for this boundary action implies that the zeta functions of the spectral triple suffice to characterize the (anti)conformal isomorphism class of the corresponding Riemann surface. Thus, you can hear
www.elsevier.com/locate/jgp Zeta functions that hear the shape of a Riemann surface
, 2007
"... To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose “Riemannian ” aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson–Sullivan ..."
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To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose “Riemannian ” aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson–Sullivan
On the distribution of orbits of geometrically finite hyperbolic groups on the boundary
, 2010
"... Abstract. We investigate the distribution of orbits of a nonelementary discrete hyperbolic subgroup Γ acting on Hn and its geometric boundary ∂∞(Hn). In particular, we show that if Γ admits a finite BowenMargulisSullivan measure (for instance, if Γ is geometrically finite), then every Γorbit i ..."
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Cited by 1 (1 self)
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orbit in ∂∞(Hn) is equidistributed with respect to the PattersonSullivan measure supported on the limit set Λ(Γ). The appendix by Maucourant is the extension of a part of his thesis where he obtains the same result as a simple application of Roblin’s theorem. Our approach is via establishing
Length distortion and the Hausdorff dimension of limit sets
 Amer. J. Math
"... Abstract. Let Γ be a convex cocompact quasiFuchsian Kleinian group. We define the distortion function along geodesic rays lying on the boundary of the convex hull of the limit set, where each ray is pointing in a randomly chosen direction. The distortion function measures the ratio of the intrinsi ..."
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Cited by 4 (0 self)
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proof of the following result of Bowen: If the limit set of Γ is not a round circle, then the Hausdorff dimension of the limit set is strictly greater than one. The proofs are developed from results in PattersonSullivan theory and ergodic theory. 1. Introduction. The
Results 1  10
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