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19
Linear representations of hereditarily nonsensitive dynamical systems
 Colloq. Math
"... Abstract. For an arbitrary topological group G any compact Gdynamical system (G, X) can be linearly Grepresented as a weak ∗compact subset of a dual Banach space V ∗. As was shown in [44] the Banach space V can be chosen to be reflexive iff the metric system (G, X) is weakly almost periodic (WAP) ..."
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Cited by 9 (9 self)
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Abstract. For an arbitrary topological group G any compact Gdynamical system (G, X) can be linearly Grepresented as a weak ∗compact subset of a dual Banach space V ∗. As was shown in [44] 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 Gsystems which can be linearly represented as a weak ∗compact subset of a dual Banach space with the RadonNikod´ym property. We call such a system a RadonNikod´ym system (RN). One of our main results is to show that for metrizable compact Gsystems 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 Gsystems such as WAP and LE (locally equicontinuous). We show that the GlasnerWeiss examples of recurrenttransitive locally equicontinuous but not weakly almost periodic cascades are actually RN. We also show that for symbolic systems the RN property is equivalent to having a countable phase space; and that any Zdynamical system (f, X), where X is either the unit interval or the unit circle and f: X → X is a homeomorphism, is an RN system. Using fragmentability and Namioka’s theorem we give an enveloping semigroup characterization of HNS and show that the enveloping semigroup of a compact metrizable HNS system is a separable Rosenthal compact, hence of cardinality ≤ 2 ℵ0. Moreover, applying a theorem of Todor˘cević we show, for discrete countable acting groups, that it admits an at most twotoone metric factor.
On tame dynamical systems
 Colloq. Math
"... Abstract. A dynamical version of the BourgainFremlinTalagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame ” topological space whose topology is determined by the convergence of sequences. In the la ..."
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Cited by 5 (1 self)
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Abstract. A dynamical version of the BourgainFremlinTalagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame ” topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame iff it is equicontinuous (ii) for an abelian acting group a tame metric minimal system is PI (hence a weakly mixing minimal system is never tame), and (iii) a tame minimal cascade has zero topological entropy. We also show that for minimal distalbutnotequicontinuous systems the canonical map from the enveloping operator semigroup onto the Ellis semigroup is never an isomorphism. This answers a long standing open question. We give a complete characterization of minimal systems whose enveloping semigroup is metrizable. In particular it follows that for abelian acting group such a system is equicontinuous.
HOMOCLINIC GROUP, IE GROUP, AND EXPANSIVE ALGEBRAIC ACTIONS
"... Abstract. We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of pexpansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1expansive exactly when it has finite ..."
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Cited by 5 (5 self)
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Abstract. We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of pexpansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclicbyfinite group on X, it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in X exactly when the action has completely positive entropy. 1.
Hereditarily nonsensitive dynamical systems and linear representations
 Colloq. Math
"... Abstract. For an arbitrary topological group G any compact Gdynamical system (G, X) can be linearly Grepresented 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) ..."
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Cited by 4 (2 self)
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Abstract. For an arbitrary topological group G any compact Gdynamical system (G, X) can be linearly Grepresented 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 Gsystems which can be linearly represented as a weak ∗compact subset of a dual Banach space with the RadonNikod´ym property. We call such a system a RadonNikod´ym system (RN). One of our main results is to show that for metrizable compact Gsystems 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 Gsystems such as WAP and LE (locally equicontinuous). We show that the GlasnerWeiss examples of recurrenttransitive 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 Gsystem is a separable Rosenthal compact, hence of cardinality ≤ 2 ℵ0. We investigate a dynamical version of the BourgainFremlinTalagrand dichotomy and a dynamical version of Todor˘cević dichotomy concerning Rosenthal compacts.
Independence in topological and C ∗ dynamics
"... Abstract. We develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods. One of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties. We also reframe our theory of dynamical ..."
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Cited by 3 (3 self)
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Abstract. We develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods. One of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties. We also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to C ∗dynamics. 1.
ALMOST PERIODICALLY FORCED CIRCLE FLOWS
"... Abstract. The present paper is a general study on dynamical and topological behaviors of minimal sets in skewproduct circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and ..."
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Cited by 2 (0 self)
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Abstract. The present paper is a general study on dynamical and topological behaviors of minimal sets in skewproduct circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere nonlocally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which are not covered by the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be LiYorke chaotic and nonalmost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2, R)valued,
Constantlength substitutions and countable scrambled sets
 Nonlinearity
"... Abstract. In this paper we provide examples of topological dynamical systems having either finite or countable scrambled sets. In particular we study conditions for the existence of LiYorke, asymptotic and distal pairs in constant–length substitution dynamical systems. Starting from a circle rotati ..."
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Cited by 2 (1 self)
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Abstract. In this paper we provide examples of topological dynamical systems having either finite or countable scrambled sets. In particular we study conditions for the existence of LiYorke, asymptotic and distal pairs in constant–length substitution dynamical systems. Starting from a circle rotation we also construct a dynamical system having Li–Yorke pairs, none of which is recurrent. 1.
Topological chaos: what may this mean
 J. Difference Equ. Appl
, 2009
"... Abstract. We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among ..."
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Cited by 1 (0 self)
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Abstract. We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others. 1.
LiYork pairs of full Hausdorff dimension for some chaotic dynamical systems
"... We show that for some simple meanwhile classical chaotic dynamical systems the set of LiYork pairs has full Hausdorff dimension on invariant sets. ..."
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We show that for some simple meanwhile classical chaotic dynamical systems the set of LiYork pairs has full Hausdorff dimension on invariant sets.
Mathematical AnalysisSlezská univerzita v Opavě
, 2008
"... Minimality, sensitivity and topological entropy in discrete dynamics Abstract of the Ph.D. Thesis ..."
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Minimality, sensitivity and topological entropy in discrete dynamics Abstract of the Ph.D. Thesis