Some recent applications of the barycenter method in geometry

Connell, Christopher; Farb, Benson

doi:10.1090/pspum/071/2024628

Cited by 10 publications

(6 citation statements)

References 29 publications

(27 reference statements)

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“…We have λ(L * n ) = (n − 1) 2 for n 4. The optimal weights are given by w(v) = 1 2 along the loop and by w(v) = 1 at the remaining vertex; thus Q(t) = 1 − nt 1/2 + (n − 1)t. 10. We have λ(L * 3 ) = 8.…”

Section: λ(H) λ(G) λ(γ)mentioning

confidence: 97%

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Entropy and the clique polynomial

McMullen

2014

Journal of Topology

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Section: λ(H) λ(G) λ(γ)mentioning

confidence: 97%

“…Constant curvature. It is known that a metric of constant negative curvature is optimal for the volume entropy on a compact manifold, given by the growth rate h(M, g) = lim T →∞ log vol(B(x, T )) 1/T of balls in its universal cover [3]; see also the surveys [4,10].…”

Section: Gradient Of Entropymentioning

confidence: 99%

Entropy and the clique polynomial

McMullen

2014

Journal of Topology

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“…The barycenter method has been employed many times, including the work of Connell and Farb on higherrank symmetric spaces [CF03a,CF03b]. See [CF03c] for a survey.…”

Section: Introductionmentioning

confidence: 99%

“…See Section 3.1.4 for the definiton, but note here that N (F ) is equal to 1 when F is Riemannian and is strictly greater than 1 otherwise. (The terminology 'eccentricity factor' is coined in [CF03c].) Their Finsler extension of Theorem 1.7 is as follows.…”

Section: Introductionmentioning

confidence: 99%

Entropy rigidity and Hilbert volume

Adeboye¹,

Bray²,

Constantine³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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For a closed, strictly convex projective manifold of dimension n ≥ 3 that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson-Courtois-Gallot's entropy rigidity result to Hilbert geometries.

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“…We also mention the entropy rigidity conjecture, which, roughly, posits that if M carries a locally symmetric metric g 0 , then h vol ðdv g 0 Þ minimizes h vol among all riemannian metrics on M with the same volume (see e.g. [2], [11]). The quantity most closely related to the volume growth rate, however, is the topological entropy of the geodesic flow.…”

Section: Introductionmentioning

confidence: 99%

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

Leuzinger

2006

Advg

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Let ðX ; d X Þ be a geodesically complete Hadamard space endowed with a Borelmeasure m. Assume that there exists a group G of isometries of X which acts totally discontinuously and cocompactly on X and preserves m. We show that the topological entropy of the geodesic flow on the space of (parametrized) geodesics of the compact quotient GnX is equal to the volume entropy of m (if X satisfies a certain local uniformity condition). This extends a result of Manning for riemannian manifolds of nonpositive curvature to the singular case. The result in particular holds for Bruhat-Tits buildings, for which we also compute the entropy explicitly.

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Some recent applications of the barycenter method in geometry

Cited by 10 publications

References 29 publications

Entropy and the clique polynomial

Entropy and the clique polynomial

Entropy rigidity and Hilbert volume

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

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