Limits of limit sets I

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon–Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometric limit is geometrically finite, the maps on limit sets converge pointwise but not uniformly.     

Slow relaxations and bifurcations of the limit sets of dynamical systems. II. Slow relaxations of a family of semiflows

We propose a number of approaches to the notion of the relaxation time of a dynamical system which are motivated by the problems of chemical kinetics, give exact mathematical definitions of slow relaxations, study their possible reasons, among which an important role is played by bifurcations of limit sets.     

On the connectedness of limit net sets

In this paper we introduce and study net sets and limit net sets. The construction and geometry of net sets can be described with the help of substitutions with net matrices which we also introduce here. Limit net sets are a special type of Moran fractals. We study connectedness properties of net sets and limit net sets.     

Wiggly sets and limit sets

We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that ifG is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1. The first author is partially supported by NSF Grant DMS 95-00577 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF grant DMS-94-23746.     

Power-law estimates for the central limit theorem for convex sets

We investigate the rate of convergence in the central limit theorem for convex sets established in [B. Klartag, A central limit theorem for convex sets, Invent. Math., in press. [8]]. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets.     

Prolongational limit sets of control systems

This article introduces the notions of prolongations and prolongational limit sets of control systems. It is shown how the prolongational limit sets play in studying recursive and dispersive concepts of control systems.     

Omega limit sets and distributional chaos on graphs

We give a full topological characterization of omega limit sets of continuous maps on graphs and we show that basic sets have similar properties as in the case of the compact interval. We also prove that the presence of distributional chaos, the existence of basic sets, and positive topological entropy (among other properties) are mutually equivalent for continuous graph maps.     

The limit sets of Schottky quasiconformal groups are uniformly perfect

In this paper we study Schottky quasiconformal groups. We show that the limit sets of Schottky quasiconformal groups are uniformly perfect, and that the limit set of a given discrete non-elementary quasiconformal group has positive Hausdorff dimension.

     

On negative limit sets for one-dimensional dynamics

In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every α-limit set is an ω-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all α-limit sets of interval maps is not closed in the Hausdorff metric (and thus some ω-limit sets are never obtained as α-limit sets). Moreover, we prove that the set of all recurrent points is closed if and only if the space of all α-limit sets is closed.     

Local and global properties of limit sets of foliations of quasigeodesic Anosov flows

A nonsingular flow is quasigeodesic when all flow lines are efficient in measuring distances in relative homotopy classes. We analyze the quasigeodesic property for Anosov flows in -manifolds which have negatively curved fundamental group. We show that this property implies that limit sets of stable and unstable leaves (in the universal cover) vary continuously in the sphere at infinity. It also follows that the union of the limit sets of all stable (or unstable) leaves is not the whole sphere at infinity. Finally, under the quasigeodesic hypothesis we completely determine when limit sets of leaves in the universal cover can intersect. This is done by determining exactly when flow lines in the universal cover share an ideal point.

     

Hausdorff dimension of the limit sets of some planar geometric constructions

We determine the Hausdorff and box dimension of the limit sets for some class of planar non-Moran-like geometric constructions generalizing the Bedford-McMullen general Sierpiński carpets. The class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines, as well as some non-affine examples, e.g. the flexed Sierpiński gasket.     

On finite limit sets for transformations on the unit interval

An infinite sequence of finite or denumerable limit sets is found for a class of many-to-one transformations of the unit interval into itself. Examples of four different types are studied in some detail; tables of numerical results are included. The limit sets are characterized by certain patterns; an algorithm for their generation is described and established. The structure and order of occurrence of these patterns is universal for the class.     

Functional central limit theorem for the volume of excursion sets generated by associated random fields

We prove a functional central limit theorem for the volume of the excursion sets generated by a stationary and associated random field with smooth realizations.     

Localization of limit sets of solutions of nonautonomous delay equations by nonmonotone functionals

We present new generalizations of the Barbashin-Krasovskii theorem which also apply to delay equations with unbounded right-hand side. These generalizations are based on information on the localization of the limit sets of solutions, which is obtained with the use of two classes of not necessarily monotone Lyapunov functionals. The classes of functionals to be used contain sign-definite Lyapunov-Krasovskii functionals as well as Lyapunov-Razumikhin functions.     

Instability conditions for solutions of delay systems and localization of limit sets

We obtain new tests for the instability of the trivial solutions of equations with deviating argument. In contrast to earlier-known results, these tests use nonmonotone Lyapunov functionals. The class of such functionals contains Lyapunov-Krasovskii functionals as well as Lyapunov-Razumikhin functions as special cases. By localizing the limit sets of solutions, in a number of instability tests, we have been able to drop the requirement that the derivative of the Lyapunov functional according to the system be negative definite.     

An explicit construction of Kleinian groups with small limit sets

This paper provides an explicit construction of Kleinian groups that have small Hausdorff dimension of their limit sets. It is known that such groups exist and they can be constructed by results of Patterson. The purpose here is to work out the methods of calculation.     

Convergence determining sets in the central limit theorem

Summary A complete characterisation is given of the class of all doublets which determine the rate of convergence in the central limit theorem. This enables a number of important properties of convergence determining sets to be deduced. In particular, it is shown that no singleton can be convergence determining, and any set consisting of four or more distinct points is convergence determining. Numerical and analytic methods are used to derive the geometry of the class of all convergence determining doublets.     

Approximation on limit compact sets of kleinian groups

Let Γ be a geometrically finite or a quasi-Fuchsian Kleiman group such that ∞ ? $\mathop \Omega \limits^o \left( v \right)$ . We establish the relation $X = clos_X L\left( {\frac{1}{{1 - a}},a \in \Xi } \right)$ for some countable sets Ξ?ω(Γ) connected with actions of elements of Γ, and for the space X=C(Γ) or for the Hölder classes X=L^α(Λ), 0<α<1, where Λ=Λ(Γ)=?\Ω is the limit set of Γ. Bibliography: 6 titles.     

Invariant currents on limit sets

We relate the L ²-cohomology of a complete hyperbolic manifold to the invariant currents on its limit set. Received: January 18, 2000