Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP_n over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FP_n is closed under extensions, quotients by subgroups of type FP_n, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP_∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FP_n but not of type FP_n+1 over Z_? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP_n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.     

Differentiability and analyticity of topological entropy for Anosov and geodesic flows

Summary In this paper we investigate the regularity of the topological entropyh _top forC ^k perturbations of Anosov flows. We show that the topological entropy varies (almost) as smoothly as the perturbation. The results in this paper, along with several related results, have been announced in [KKPW].Partially supported by NSF Grant DMS85-14630     

On the topological equivalence of some fuzzy flows near hyperbolic equilibria

The purpose of this work is to study fuzzy dynamical systems associated with deterministic systems. The Grobman-Hartman theorem states that, near hyperbolic equilibria, there exists a homeomorphism between the nonlinear system's trajectories and the linearized correspondent system's trajectories. That is, these systems are topologically equivalent. A theorem similar to Grobman-Hartman theorem to fuzzy flows is the main result in this article. For fuzzy flows obtained from each system it states that the nonlinear and the linearized are topologically equivalent.     

Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds

Let be the total space of a fibre bundle with base a simply connected manifold whose loop space homology grows exponentially for a given coefficient field. Then we show that for any Riemannian metric on , the topological entropy of the geodesic flow of is positive. It follows then, that there exist closed manifolds with arbitrary fundamental group, for which the geodesic flow of any Riemannian metric on has positive topological entropy.

     

Lowering topological entropy

The main result we prove in this paper is that for any finite dimensional dynamical system (with topological entropyh), and for any factor with strictly lower entropyh′, there exists an intermediate factor of entropyh″ for everyh″∈[h′, h]. Two examples, one of them minimal, show that this is not the case for infinite dimensional systems.     

Topology and topological sequence entropy

Let X be a compact metric space and T:X-→X be continuous.Let h*(T)be the supremum of topological sequence entropies of T over all the subsequences of Z+and S(X)be the set of the values h*(T)for all the continuous maps T on X.It is known that{0}■S(X)■{0,log 2,log 3,...}∪{∞}.Only three possibilities for S(X)have been observed so far,namely S(X)={0},S(X)={0,log 2,∞}and S(X)={0,log 2,log 3,...}∪{∞}.In this paper we completely solve the problem of finding all possibilities for S(X)by showing that in fact for every set{0}?A?{0,log 2,log 3,...}∪{∞}there exists a one-dimensional continuum XAwith S(XA)=A.In the construction of XAwe use Cook continua.This is apparently the first application of these very rigid continua in dynamics.We further show that the same result is true if one considers only homeomorphisms rather than continuous maps.The problem for group actions is also addressed.For some class of group actions(by homeomorphisms)we provide an analogous result,but in full generality this problem remains open.     

Local audibility of a hyperbolic metric

A Riemannian metric g on a compact boundaryless manifold is said to be locally audible if the following statement is true for every metric g′ sufficiently close to g: if g and g′ are isospectral then they are isometric. The local audibility is proved of a metric of constant negative sectional curvature.     

A topological characterisation of hyperbolic groups

We characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space. This is, in turn, equivalent to a convergence group for which every point of the space is a conical limit point.

     

Partially hyperbolic geodesic flows

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.     

On the relation between topological entropy and entropy dimension

Mathematical Notes - For the Lipschitz mapping of a metric compact set into itself, there is a classical upper bound on topological entropy, namely, the product of the entropy dimension of the...     

Regularity of quasi-minimizers on metric spaces

Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasi-minimizers, and in particular p-harmonic functions, satisfy Harnack's inequality, the strong maximum principle, and are locally H?lder continuous, if the space is doubling and supports a Poincaré inequality. Received: 12 May 2000 / Revised version: 20 April 2001     

On topological and metric critical point theory

Starting from the concept of Morse critical point, introduced in [19], we propose a possible approach to critical point theory for continuous functionals defined on topological spaces, which includes some classical results, also in an infinite-dimensional setting.     

Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.     

Minimal self-joinings and positive topological entropy

We show that the properties of almost minimal self-joinings and strong almost minimal self-joinings, introduced by del Junco in Topological Dynamics, are compatible with positive topological entropy, as opposed to the stronger property of minimal self-joinings. This is done both by proving existence theorems and by explicitly constructing some symbolic systems having these properties, which are modifications of the Chacón system. It is shown furthermore that these systems have no non-trivial factors with completely positive topological entropy.     

Diffeomorphisms with positive metric entropy

We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.     

Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.     

Hyperbolic metric,curvature of geodesics and hyperbolic discs in hyperbolic plane domains

The purpose of this paper is to investigate the relations among some geometric quantities defined for every hyperbolic plane domain of any connectivity, each of which measures, in some sense, how much the domain deviates either from a disc, convex domain, or simply connected domain on one hand, or a punctured domain on the other hand. Supported by the Landau Center for Mathematical Research in Analysis.