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.     

On upper estimates for the hausdorff dimension of negatively invariant sets of local cocycles

The paper is concerned with negatively invariant sets of local cocycles generated, in particular, by nonautonomous ordinary differential equations. Upper estimates for the Hausdorff dimension for negatively invariant sets of local cocycles are obtained using singular numbers of linearization of the cocycle and special functions of Lyapunov type.     

Ricci flow of negatively curved incomplete surfaces

We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of well-posedness in this class.     

Ends of negatively curved surfaces in Euclidean space

We examine the geometry of a complete, negatively curved surface isometrically embedded in ${\mathbb{R}^3}$ . We are especially interested in the behavior of the ends of the surface and its limit set at infinity. Various constructions are developed, and a classification theorem is obtained, showing that every possible end type for a topologically finite surface with at least one bowl end arises, as well as all infinite type surfaces with a single nonannular end. Some other examples are given with oddly behaved bowl ends.     

The exact hausdorff dimension of a branching set

Summary We obtain a critical function for which the Hausdorff measure of a branching set generated by a simple Galton-Watson process is positive and finite.     

The hausdorff dimension of the sample path of a subordinator

The Hausdorff dimension of the range of an arbitrary subordinator is exactly determined in terms of the rate of linear drift and the Levy measure of the subordinator. This generalizes the result of Blumenthal and Getoor: that for a stable subordinator of indexσ, the dimension of the range isσ.     

Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

In this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface.     

A lower bound for Hausdorff dimensions of harmonic measures on negatively curved manifolds

Employing the methods of [KL], a lower bound for Hausdorff dimension of harmonic measures on negatively curved manifolds is derived yielding, in particular, that if the curvature tends to a constant then the above Hausdorff dimension tends to the dimension of the sphere at infinity. Supported by U.S.-Israel BSF.     

On hausdorff dimension of some cantor attractors

We study what happens with the dimension of Feigenbaum-like attractors of smooth unimodal maps as the order of the critical point grows.     

Continuity of the hausdorff dimension for piecewise monotonic maps

In this paper a piecewise monotonic mapT:X→?, whereX is a finite union of intervals, is considered. DefineR(T)= \(\mathop \cap \limits_{n = 0}^\infty \overline {T^{ - n} X} \) . The influence of small perturbations ofT on the Hausdorff dimension HD(R(T)) ofR(T) is investigated. It is shown, that HD(R(T)) is lower semi-continuous, and an upper bound of the jumps up is given. Furthermore a similar result is shown for the topological pressure.     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

Unclouding the sky of negatively curved manifolds

Let M be a complete simply connected Riemannian manifold, with sectional curvature K ≤ −1. Under certain assumptions on the geometry of ∂M, which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M, and any point x₀ outside these horoballs, it is possible to shrink uniformly, by a finite amount depending only on M, these horoballs so that some geodesic ray starting from x₀ avoids the shrunk horoballs. As an application, we give a uniform upper bound on the infimum of the heights of the closed geodesics in the finite volume quotients of M.Received: January 2004 Accepted: August 2004     

Growth tightness of negatively curved manifolds

We show that any closed negatively curved manifold X is growth tight: this means that its universal covering $\text{X}$ has an exponential growth rate $\text{ω(}\text{X}\text{)}$ which is strictly greater than the exponential growth rate $\text{ω(}\text{X}\text{)}$ of any other normal covering $\text{X}$. Moreover, we give an explicit formula which estimates the difference between $\text{ω(}\text{X}\text{)}$ and $\text{ω(}\text{X}\text{)}$ in terms of the systole of $\text{X}$ and of some geometric parameters of the base manifold X. Then, we describe some applications to systoles and periodic geodesics. To cite this article: A. Sambusetti, C. R. Acad. Sci. Paris, Ser. I 336 (2003).