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.

Information geometry of the space of probability measures and barycenter maps

2021 ◽  
Vol 34 (2) ◽  
pp. 231-253
Author(s):  
Mitsuhiro Itoh ◽  
Hiroyasu Satoh
Keyword(s):  
Information Geometry ◽  
Hadamard Manifold ◽  
Probability Measures ◽  
Ideal Boundary ◽  
Recent Developments ◽  
Fisher Metric ◽  
The Ideal

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures, and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.

Time-preserving conjugacies of geodesic flows

1992 ◽  
Vol 12 (1) ◽  
pp. 67-74 ◽  
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Sufficient Conditions ◽  
Universal Covering ◽  
Probability Measures ◽  
Geodesic Flows ◽  
Ideal Boundary ◽  
Negatively Curved ◽  
Unit Tangent Bundle ◽  
Borel Probability ◽  
Necessary And Sufficient ◽  
The Ideal

AbstractIn this note we study Borel-probability measures on the unit tangent bundle ofa compact negatively curved manifold M that are invariant under the geodesic flow. We interpret the entropy of such a measure as a Hausdorff dimension with respect to a natural family of distances on the ideal boundary of the universal covering of M. This in term yields necessary and sufficient conditions for the existence of time preserving conjugacies of geodesic flows.

scholarly journals The ideal boundaries and global geometric properties of complete open surfaces

1990 ◽  
Vol 120 ◽  
pp. 181-204 ◽  
Author(s):  
Takashi Shioya
Keyword(s):  
Equivalence Relation ◽  
Focal Point ◽  
Asymptotic Relation ◽  
Total Curvature ◽  
Equivalence Classes ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Ideal Boundary ◽  
Geometric Properties ◽  
The Ideal

In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.

Polynomial invariants for virtual marked graphs and virtual surface-link theory I

2017 ◽  
Vol 26 (12) ◽  
pp. 1750081
Author(s):  
Sang Youl Lee
Keyword(s):  
Natural Extension ◽  
Polynomial Invariants ◽  
Kauffman Bracket ◽  
Virtual Links ◽  
Link Theory ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Ideal

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

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.

scholarly journals Order-Invariant Measures on Fixed Causal Sets

2012 ◽  
Vol 21 (3) ◽  
pp. 330-357 ◽  
Author(s):  
GRAHAM BRIGHTWELL ◽  
MALWINA LUCZAK
Keyword(s):  
Linear Extension ◽  
Invariant Measures ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Order Type ◽  
Probability Measures ◽  
Causal Sets ◽  
Natural Extensions ◽  
Countably Infinite ◽  
Order Invariant

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers; we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each feasible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

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.

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