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 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

Local rigidity of Anosov higher-rank lattice actions

1998 ◽  
Vol 18 (3) ◽  
pp. 687-702 ◽  
Author(s):  
NANTIAN QIAN ◽  
CHENGBO YUE
Keyword(s):  
Abelian Group ◽  
Lyapunov Functional ◽  
Rigidity Result ◽  
Absolute Value ◽  
Local Rigidity ◽  
Higher Rank ◽  
Standard Action ◽  
Lattice Actions

Let $\rho_0$ be the standard action of a higher-rank lattice $\Gamma$ on a torus by automorphisms induced by a homomorphism $\pi_0:\Gamma\to SL(n,{\Bbb Z})$. Assume that there exists an abelian group ${\cal A}\subset \Gamma$ such that $\pi_0({\cal A})$ satisfies the following conditions: (1) ${\cal A}$ is ${\Bbb R}$-diagonalizable; (2) there exists an element $a\in {\cal A}$, such that none of its eigenvalues $\lambda_1,\dots,\lambda_n$ has unit absolute value, and for all $i,j,k=1,\dots,n$, $|\lambda_i\lambda_j|\neq|\lambda_k|$; (3) for each Lyapunov functional $\chi_i$, there exist finitely many elements $a_j\in {\cal A}$ such that $E_{\chi_i}=\cap_{j} E^u(a_j)$ (see \S1 for definitions). Then $\rho_0$ is locally rigid. This local rigidity result differs from earlier ones in that it does not require a certain one-dimensionality condition.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

2016 ◽  
Vol 27 (10) ◽  
pp. 1650085
Author(s):  
A. Baklouti ◽  
N. Elaloui ◽  
I. Kedim
Keyword(s):  
Homogeneous Space ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Rigidity Theorem ◽  
General Context ◽  
Affine Transformations ◽  
Discontinuous Group ◽  
Solvable Case ◽  
Local Rigidity ◽  
Discontinuous Groups

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

Right ventricular outflow tract obstruction: a quest for ideal management

2018 ◽  
Vol 26 (6) ◽  
pp. 451-460 ◽  
Author(s):  
John SK Murala ◽  
Ryan J Vela ◽  
Tracy Geoffrion ◽  
Surpreet Chopra ◽  
Soma Guhathakurtha ◽  
...  
Keyword(s):  
Genetic Engineering ◽  
Right Ventricular ◽  
Right Ventricular Outflow Tract ◽  
Ventricular Outflow Tract ◽  
Outflow Tract ◽  
New Techniques ◽  
Ongoing Research ◽  
Ventricular Outflow Tract Obstruction ◽  
The Ideal

Management of right ventricular outflow tract obstruction has undergone much change over the last century. Techniques described in the literature include anatomical repairs and the use of various patches, conduits, and innovative grafts. However, many of these approaches require reoperations or catheter-based interventions, leading to increased morbidity, mortality, and cost. The search for the ideal long-lasting conduit continues and there are new techniques on the horizon, using genetic engineering and nanotechnology. This review discusses the evolution of various techniques for repair of right ventricular outflow tract obstruction, past and current conduits, as well as ongoing research.

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.

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