scholarly journals Behavior of Eigenfunctions near the Ideal Boundary of Hyperbolic Space

2018 ◽  
Author(s):  
Harold Donnelly
Keyword(s):  
Hyperbolic Space ◽  
Ideal Boundary ◽  
The Ideal

scholarly journals A Riemann on the ideal boundary of a Riemann surface

1956 ◽  
Vol 32 (6) ◽  
pp. 409-411 ◽  
Author(s):  
Shin'ichi Mori ◽  
Minoru Ota
Keyword(s):  
Riemann Surface ◽  
Ideal Boundary ◽  
The Ideal

The parabolicity of Brelot's harmonic spaces

1995 ◽  
Vol 59 (1) ◽  
pp. 20-27
Author(s):  
Hideo Imai
Keyword(s):  
Harmonic Function ◽  
Ideal Boundary ◽  
Minimal Growth ◽  
Positive Harmonic Function ◽  
The Ideal

AbstractThe parabolicity of Brelot's harmonic spaces is characterized by the fact that every positive harmonic function is of minimal growth at the ideal boundary.

Information geometry of Busemann-barycenter for probability measures

2015 ◽  
Vol 26 (06) ◽  
pp. 1541007 ◽  
Author(s):  
Mitsuhiro Itoh ◽  
Hiroyasu Satoh
Keyword(s):  
Natural Extension ◽  
Information Geometry ◽  
Acta Math ◽  
Space Structure ◽  
Hadamard Manifold ◽  
Probability Measures ◽  
Positive Density ◽  
Ideal Boundary ◽  
Push Forward ◽  
The Ideal

Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫+(∂X, dθ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫+(∂X, dθ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de coubure strictement négative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.

scholarly journals Carnot metrics, dynamics and local rigidity

2021 ◽  
pp. 1-51
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Symmetric Spaces ◽  
Holonomy Group ◽  
Tangent Cones ◽  
Ideal Boundary ◽  
Uniform Lattice ◽  
New Techniques ◽  
Carathéodory Metric ◽  
Local Rigidity ◽  
Lattice Actions ◽  
The Ideal

Abstract This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

scholarly journals Radon-Nikodym Densities between Harmonic Measures on the Ideal Boundary of an Open Riemann Surface

1966 ◽  
Vol 27 (1) ◽  
pp. 71-76
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Compact Set ◽  
Ideal Boundary ◽  
Subharmonic Functions ◽  
Open Riemann Surface ◽  
Harmonic Measures ◽  
The Ideal

Resolutive compactification and harmonic measures. Let R be an open Riemann surface. A compact Hausdorff space R* containing R as its dense subspace is called a compactification of R and the compact set Δ = R* -R is called an ideal boundary of R. Hereafter we always assume that R does not belong to the class OG. Given a real-valued function f on Δ, we denote by the totality of lower bounded superharmonic (resp. upper bounded subharmonic) functions sonis satisfying

The physical states of a boundary conformal field theory

2020 ◽  
Vol 35 (06) ◽  
pp. 2050021
Author(s):  
Simon Davis
Keyword(s):  
Field Theory ◽  
Hilbert Space ◽  
Conformal Field Theory ◽  
Vacuum State ◽  
Linear Measure ◽  
Ideal Boundary ◽  
Residue Theorem ◽  
Conformal Field ◽  
Nonperturbative Effect ◽  
The Ideal

The path integral of a conformal field theory on a bordered Riemann surface defines a state in a Hilbert space on this boundary. Over the ideal boundary, the Hausdorff dimension may be less than one. The integral representing the flux over the ideal boundary is evaluated through a generalization of the residue theorem. The identification of the state for infinite-genus surfaces with the vacuum state with a perturbative vacuum is distinguished from the Hilbert space on ideal boundaries of nonzero linear measure. This nonperturbative effect is identified as an instanton in a separate quantum theory.

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