Hyperbolicity Research Articles

The Poisson Formula for Groups with Hyperbolic Properties

The Poisson boundary of a group G with a probability measure „ is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded „-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with inflnitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semisimple Lie groups.

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT2, we show that the set of Riemannian metrics inT2 with no geodesic foliations having rotation numberh isCk dense for everyk ∈ N. We also show that, generically in theC2 topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.

A note on stable exponents

Given an element γ in a group γ, the stable exponent p+(γ) of γ is defined as p+(γ) =lim supn→∞P(γn) denotes the exponent of P(γn) = sup{k/ ∀γo ∈ γ such that γn = γk o We prove that if γ acts properly discontinuously by isometrics on a proper geodesic Gromov-hyperbolic metric space and γ ∈ γ is of hyperbolic type, then P+(γ) is an integer. This implies that the stable exponent of every element of infinite order in a word hyperbolic group is an integer. We also show that, in a translation discrete group, the stable exponent of every element of infinite order is finite.

The mapping class group and a theorem of Masur–Wolf

In (Masur and Wolf, 1995), Masur and Wolf proved that the Teichmüller space of genus g>1 surfaces with the Teichmüller metric is not a Gromov hyperbolic space. In this paper, we provide an alternative proof based upon a study of the action of the mapping class group on Teichmüller space.

Topological stability and Gromov hyperbolicity

We show that if the geodesic flow of a compact analytic Riemannian manifold $M$ of non-positive curvature is either $C^{k}$-topologically stable or satisfies the $\epsilon$-$C^{k}$-shadowing property for some $k > 0$ then the universal covering of $M$ is a Gromov hyperbolic space. The same holds for compact surfaces without conjugate points.

Tautness and complete hyperbolicity of domains in $\mathbb {C}^n$

Tautness and complete hyperbolicity of domains in $\mathbb {C}^n$

Hyperbolic framework for thermoelastic materials

A THERMODYNAMIC framework of a deformable continuum is developed in which the conservative state variable vector is enlarged by adding the spatial gradient of a scalar thermal internal state variable responsible for the description of thermal history effects. The theory leads to a modified model of thermoelasticity with internal state variables and with a wave-type heat conduction governed by a system of quasi-linear hyperbolic equations. In a general non-deformable case, the observed material properties such as specific heat, quasi-equilibrium thermal conductivity and speed of thermal (the so-called second sound) waves, all regarded as functions of , lead to a particular specification of all material functions and the evolution equation for the scalar internal state variable. The short review of the heat pulse experiment is made. Main assumptions of the present approach are formulated and some arguments referred to the rate-type description are presented. A set of remarks and comparisons with another modification of the Fourier law characterizes the model. An analysis of hyperbolicity of thermoelasticity by propagation conditions of weak discontinuity waves is performed.

The Second Bounded Cohomology of a Group Acting on a Gromov-Hyperbolic Space

Suppose a group G acts on a Gromov-hyperbolic space X properly discontinuously. If the limit set L(G) of the action has at least three points, then the second bounded cohomology group of G, H b 2 ( G ; R ) is infinite dimensional. For example, if M is a complete, pinched negatively curved Riemannian manifold with finite volume, then H b 2 ( π 1 ( M ) ; R ) is infinite dimensional. As an application, we show that if G is a knot group with G≄Z, then H b 2 ( G ; R ) is infinite dimensional. 1991 Mathematics Subject Classification: primary 20F32; secondary 53C20, 57M25.

Generic Hyperbolicity in the Logistic Family

Generic Hyperbolicity in the Logistic Family

Quasi-geodesic segments and Gromov hyperbolic spaces

It is known that for a geodesic metric space hyperbolicity in the sense of Gromov implies geodesic stability. In this paper it is shown that the converse is also true. So Gromov hyperbolicity and geodesic stability are equialent for geodesic metric spaces.

Boundary dimension in negatively curved spaces

LetX be a negatively curved (Gromov hyperbolic) space. We construct a bound on dim ∂X when a group of isometries acts cocompactly onX. We construct an example of a negatively curved space with infinite-dimensional boundary.

Abundance of hyperbolicity in the $C^1$ topology

— In this paper we show that the Newhouse's thickness of basic sets as an obstmction to the density of Axiom A diffeomorphisms in the C^ topology for k > 2 is not applicable in the C topology. We prove that the local thickness of saddle-type hyperbolic basic sets vanishes C generically and that, when we consider families with first homoclinic bifurcations, the ^-stability is a phenomena after the unfolding of the tangency. RESUME. — Dans cet article on montre que 1'epaisseur de Newhouse des ensembles basiques telle qu'une obstruction pour la densite des diffeomorphismes Axiome A dans la topologie (7^, k > 2, n'est pas applicable dans la topologie C. On montre que 1'epaisseur locale des ensembles basiques hyperboliques de type selle s'annule generiquement dans la topologie C et que, si 1'on considere des families exhibant une premiere bifurcation homoclinique, l'f2-stabilite est un phenomene « prevalent » apres Ie developpement de la tangence.

Hyperbolicity of localizations

Hyperbolicity of localizations

Families of curves and hyperbolicity

Families of curves and hyperbolicity

Leafwise hyperbolicity; a correction

Leafwise hyperbolicity; a correction

Hyperbolicity and branched coverings

Hyperbolicity and branched coverings

A branched covering of CP2?S 4, hyperbolicity and projectivity topology

A branched covering of CP2?S 4, hyperbolicity and projectivity topology

Generic hyperbolicity in one-dimensional reaction- diffusion equations with general boundary conditions

On considere l'equation u t =u xx +f(u), 0 0, f∈C 1 (R,R) sous les conditions aux limites: p(u x ,(0,t))+g 0 (u(0,t))+(1-p)u(0,t)=p(u x (1,t))+g 1 (u(1,t))+(1-p)u(1,t)=0 ou p∈{0,1} et g i ∈( 1 (R,R). On demontre que generiquement toutes les solutions stationnaires sont hyperboliques

Some theorems on projective hyperbolicity

Some theorems on projective hyperbolicity

On the nation of hyperbolicity

Communications on Pure and Applied MathematicsVolume 33, Issue 3 p. 395-397 Article On the nation of hyperbolicity† Peter D. Lax, Peter D. LaxSearch for more papers by this author Peter D. Lax, Peter D. LaxSearch for more papers by this author First published: May 1980 https://doi.org/10.1002/cpa.3160330309Citations: 16 † Dedicated to Fritz John on the Occasion of his 70th Birthday, June 14, 1980 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume33, Issue3May 1980Pages 395-397 RelatedInformation