Extended Dynamical Systems and Stability Theory

In this paper we characterize tame dynamical systems and functions in terms of eventual non-sensitivity and eventual fragmentability. As a notable application we obtain a neat characterization of tame subshifts X ⊂ {0, 1} Z : for every infinite subset L ⊆ Z there exists an infinite subset K ⊆ L such that π K (X) is a countable subset of {0, 1} K . The notion of eventual fragmentability is one of the properties we encounter which indicate some "smallness" of a family. We investigate a "smallness hierarchy" for families of continuous functions on compact dynamical systems, and link the existence of a "small" family which separates points of a dynamical system (G, X) to the representability of X on "good" Banach spaces. For example, for metric dynamical systems the property of admitting a separating family which is eventually fragmented is equivalent to being tame. We give some sufficient conditions for coding functions to be tame and, among other applications, show that certain multidimensional analogues of Sturmian sequences are tame. We also show that linearly ordered dynamical systems are tame and discuss examples where some universal dynamical systems associated with certain Polish groups are tame.

1998

2012

The first measure of noncompactness was defined by Kuratowski in 1930 and later the Hausdorff measure of noncompactness was introduced in 1957 by Goldenštein et al. These measures of noncompactness have various applications in several areas of analysis, for example, in operator theory, fixed point theory, and in differential and integral equations.

Colloquium Mathematicum, 2006

For an arbitrary topological group G any compact G-dynamical system (G, X) can be linearly G-represented as a weak * -compact subset of a dual Banach space V * . As was shown in [45] the Banach space V can be chosen to be reflexive iff the metric system (G, X) is weakly almost periodic (WAP). In this paper we study the wider class of compact G-systems which can be linearly represented as a weak * -compact subset of a dual Banach space with the Radon-Nikodým property. We call such a system a Radon-Nikodým system (RN). One of our main results is to show that for metrizable compact G-systems the three classes: RN, HNS (hereditarily not sensitive) and HAE (hereditarily almost equicontinuous) coincide. We investigate these classes and their relation to previously studied classes of G-systems such as WAP and LE (locally equicontinuous). We show that the Glasner-Weiss examples of recurrent-transitive locally equicontinuous but not weakly almost periodic cascades are actually RN. Using fragmentability and Namioka's theorem we give an enveloping semigroup characterization of HNS systems and show that the enveloping semigroup E(X) of a compact metrizable HNS G-system is a separable Rosenthal compact, hence of cardinality ≤ 2 ℵ 0 . We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of Todorcević dichotomy concerning Rosenthal compacts. Of course the classical theory of WAP functions is valid for a general topological group G and it is not hard to see that the AE theory, as well as the theory of K(G)-functionswhich we call Veech functions -extend to such groups as well.

impulsive di#erential equations, Descartes Press, Koriyama, 1996. [16]P. W. Bates and K. Lu, A Hartman--Grobman theorem for Cahn--Hillard equations and phases--field equations, J. Dynam. Di#erential Equations 6 (1994), no. 1, 101--145. [17]N. N. Bogolyubov, On some statistical methods in mathematical physics, AN USSR, L'vov, 1945 (Russian). [18]N. N. Bogolyubov and Ju. A. Mitropol'ski, Asymptotic methods in the theory of nonlinear oscillations, Gordon and Breach, New York, 1962. [19]P. Bohl, Uber die Bewegung eines mechanischen Systems in der Nahe einer Gleichgewichtslage, J. Reine Angew. Math. 127 (1904), no. 3--4, 179--276. [20]M. A. Boudourides, Topological equivalence of monotone nonlinear nonautonomous di#erential equations, Portugal. Math. 41 (1982), no. 1--4, 287--294. [21]M. A. Boudourides, Hyperbolic Lipschitz homeomorphisms and flows, in Equadi# 82, Proceedings 82, Lecture Notes in Math., 1017, 1983, pp. 101--106. [22]R. Bowen, #--limit sets for Axiom A di#eomorphi...