Geometrically finite groups,Patterson-Sullivan measures and Ratner's ridigity theorem

Summary The rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated. Ratner's theorem generalizes to these foliations with respect to the Patterson-Sullivan measure. In the spirit of Mostow, we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups. Finally, we show that the frame flow on geometrically finite hyperbolic manifolds is Bernoulli.partially supported by NSF Grant No. DMS-820-04024partially supported by NSF Grant No. DMS-85-02319     

Ergodicity of generalised Patterson-Sullivan measures in higher rank symmetric spaces

Let X = G/K be a higher rank symmetric space of noncompact type and a discrete Zariski dense group. In a previous article, we constructed for each G-invariant subset of the regular limit set of Γ a family of measures, the so-called (b, Γ · ξ)-densities. Our main result here states that these densities are Γ-ergodic with respect to an important subset of the limit set which we choose to call the ``ray limit set'. In the particular case of uniform lattices and products of convex cocompact groups acting on the product of rank one symmetric spaces every limit point belongs to the ray limit set, hence our result is most powerful for these examples. For nonuniform lattices, however, it is a priori not clear whether the ray limit set has positive measure with respect to a (b, Γ · ξ)-density. Using a counting theorem of Eskin and McMullen, we are able to prove that the ray limit set has full measure in each G-invariant subset of the limit set.     

Elliott-Morse measures and Kakutani's dichotomy theorem

Dedicated to the memory of John Oxtoby     

A dichotomy property for Markov measures

Academy of Economic Studies, Department of Economics Cybernetics, Mathematics Chair, Calea Dorobariti 15–17, 71131 Bucharest, Romania. Published in Lietuvos Matematikos Rinkinys, Vol. 34, No. 4, pp. 533–537, October–December, 1994.     

A dichotomy for the convex spaces of probability measures

We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has ‘small’ local character in M or else M contains a measure of ‘large’ Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions.     

Projections of orbital measures for classical Lie groups

In this paper we compute the radial parts of the projections of orbital measures for the compact Lie groups of B, C, and D type, extending previous results obtained for the case of the unitary group by Olshanski and Faraut. Applying the method of Faraut, we show that the radial part of the projection of an orbital measure is expressed in terms of a B-spline with knots located symmetrically with respect to zero.     

On the Patterson-Sullivan measure of some discrete group of isometries

In this paper, we describe a large class of groups of isometries of thed-dimensional hyperbolic space. These groups may be non-geometrically finite but their Patterson-Sullivan measure is always finite.     

A dichotomy result about Hessenberg matrices associated with measures in the unit circle

We characterize Hessenberg matrices D associated with measures in the unit circle ν, which are matrix representations of compact and actually Hilbert Schmidt perturbations of the forward shift operator as those with recursion coefficients verifying , ie, associated with measures verifying Szegö condition. As a consequence, we obtain the following dichotomy result for Hessenberg matrices associated with measures in the unit circle: either D = S _R+ K ₂ with K ₂, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that with K ₂, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if , then D = S _R+ K _p with K _p an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle.     

Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ={Φ(m,n)}_(m,n)∈Δ the subspace . Supposing that X₁ is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on X₁, we obtain that the admissibility of the pair with p∈(1,∞] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.     

Exponential dichotomy and dichotomy radius for difference equations

The aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces. We classify the classes of input and output spaces, respectively, and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems. By an example we show that the obtained results are the most general in this topic. As an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input-output operators acting on sequence spaces which are invariant under translations.     

L-spaces and the P-ideal dichotomy

We extend a theorem of Todor?evi?: Under the assumption ( $ \mathcal{K} $ ) (see Definition 1.11), $$ \boxtimes \left\{ \begin{gathered} any regular space Z with countable tightness such that \hfill \\ Z^n is Lindel\ddot of for all n \in \omega has no L - subspace. \hfill \\ \end{gathered} \right. $$ We assume $ \mathfrak{p} $ > ω ₁ and a weak form of Abraham and Todor?evi?’s P-ideal dichotomy instead and get the same conclusion. Then we show that $ \mathfrak{p} $ > ω ₁ and the dichotomy principle for P-ideals that have at most ?₁ generators together with ? do not imply that every Aronszajn tree is special, and hence do not imply (ie1-4). So we really extended the mentioned theorem.     

Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets

We extend the part of Patterson-Sullivan theory to discrete quasiconformal groups that relates the exponent of convergence of the Poincaré series to the Hausdorff dimension of the limit set. In doing so we define new bi-Lipschitz invariants that localize both the exponent of convergence and the Hausdorff dimension. We find these invariants help to expose and explain the discrepancy between the conformal and quasiconformal setting of Patterson-Sullivan theory.

     

The Decay Rate of Patterson-Sullivan Measures with Potential Functions and Critical Exponents

Basing upon the recent development of the Patterson-Sullivan measures with a Hölder continuous nonzero potential function, we use tools of both dynamics of geodesic flows and geometric properties of negatively curved manifolds to present a new formula illustrating the relation between the exponential decay rate of Patterson-Sullivan measures with a Hölder continuous potential function and the corresponding critical exponent.     

On the dichotomy of Perron numbers and beta-conjugates

Let β > 1 be an algebraic number. A general definition of a beta-conjugate of β is proposed with respect to the analytical function ${f_{\beta}(z) =-1 + \sum_{i \geq 1} t_i z^i}$ associated with the Rényi β-expansion d _β (1) = 0.t ₁ t ₂ . . . of unity. From Szeg?’s Theorem, we study the dichotomy problem for f _β (z), in particular for β a Perron number: whether it is a rational fraction or admits the unit circle as natural boundary. The first case of dichotomy meets Boyd’s works. We introduce the study of the geometry of the beta-conjugates with respect to that of the Galois conjugates by means of the Erd?s–Turán approach and take examples of Pisot, Salem and Perron numbers which are Parry numbers to illustrate it. We discuss the possible existence of an infinite number of beta-conjugates and conjecture that all real algebraic numbers > 1, in particular Perron numbers, are in ${{\rm C}_1 \cup \,{\rm C}_2 \cup \,{\rm C}_3}$ after the classification of Blanchard/Bertrand-Mathis.     

指数型二分性与不变流形

该文证明了非线性微分方程组在对应的齐次方程具有指数型二分性、非线性部分满足适当的条件下存在稳定流形和不稳定流形；并且对所得的结果给出一个应用．     

凸性与度量投影的连续性

本文研究近强凸、近非常凸Banach空间中度量投影的连续性。获得如下结果：若A是近强凸（近非常凸）空间中的逼近凸集,则度量投影PA是范-范上半连续的（范-弱上半连续的）。此外,我们还利用度量投影的连续性给出Banach空间为近强凸、近非常凸的一些充分必要条件。     

A note on the dichotomy spectrum

In many ways, exponential dichotomies are an appropriate hyperbolicity notion for nonautonomous linear differential or difference equations. The corresponding dichotomy spectrum generalizes the classical set of eigenvalues or Floquet multipliers and is therefore of eminent importance in a stability theory for explicitly time-dependent systems, as well as to establish a geometric theory of nonautonomous problems with ingredients like invariant manifolds and normal forms, or to deduce continuation and bifurcation techniques.

In this note, we derive some invariance and perturbation properties of the dichotomy spectrum for nonautonomous linear difference equations in Banach spaces. They easily follow from the observation that the dichotomy spectrum is strongly related to a weighted shift operator on an ambient sequence space.     

A note on the dichotomy spectrum

We point out an error in the above short note and correct it under additional assumptions.