On local aspects of topological weak mixing, sequence entropy and chaos

2013 ◽  
Vol 34 (5) ◽  
pp. 1615-1639 ◽  
Author(s):  
PIOTR OPROCHA ◽  
GUOHUA ZHANG
Keyword(s):  
Dynamical Systems ◽  
Qualitative Theory ◽  
Broad Sense ◽  
Weak Mixing ◽  
Sequence Entropy ◽  
Weakly Mixing ◽  
Topological Weak Mixing ◽  
Topological Sequence Entropy ◽  
Mixing Sequence

AbstractIn this paper we show that for every$n\geq 2$there are minimal systems with perfect weakly mixing sets of order$n$and all weakly mixing sets of order$n+ 1$trivial. We present some relations between weakly mixing sets and topological sequence entropy; in particular, we prove that invertible minimal systems with non-trivial weakly mixing sets of order three always have positive topological sequence entropy. We also study relations between weak mixing of sets and other well-established notions from qualitative theory of dynamical systems like (regional) proximality, chaos and equicontinuity in a broad sense.

scholarly journals ERGODIC PROPERTIES OF BOGOLIUBOV AUTOMORPHISMS IN FREE PROBABILITY

2010 ◽  
Vol 13 (03) ◽  
pp. 393-411 ◽  
Author(s):  
FRANCESCO FIDALEO ◽  
FARRUKH MUKHAMEDOV
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Free Probability ◽  
Classical System ◽  
Weak Mixing ◽  
Von Neumann ◽  
Commutation Relations ◽  
Ergodic Properties ◽  
Weakly Mixing ◽  
Fixed Point Algebra

We show that some C*-dynamical systems obtained by free Fock quantization of classical ones, enjoy ergodic properties much stronger than their boson or fermion analogous. Namely, if the classical dynamical system (X, T, μ) is ergodic but not weakly mixing, then the resulting free quantized system (𝔊, α) is uniquely ergodic (w.r.t. the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system (X, T, μ) which is weakly mixing but not mixing. In this case, the free quantized system is uniquely weak mixing but not uniquely mixing. Finally, a free quantized system arising from a classical mixing dynamical system, will be uniquely mixing. In such a way, it is possible to exhibit uniquely weak mixing and uniquely mixing C*-dynamical systems whose Gelfand–Naimark–Segal representation associated to the unique invariant state generates a von Neumann factor of one of the following types: I∞, II1, IIIλwhere λ ∈ (0, 1]. The resulting scenario is then quite different from the classical one. In fact, if a classical system is uniquely mixing, it is conjugate to the trivial one consisting of a singleton. For the sake of completeness, the results listed above are extended to the q-Commutation Relations, provided [Formula: see text]. The last result has a self-contained meaning as we prove that the involved C*-dynamical systems based on the q-Commutation Relations are conjugate to the corresponding one arising from the free case (i.e. q = 0), at least if [Formula: see text].

scholarly journals On weak mixing, minimality and weak disjointness of all iterates

2011 ◽  
Vol 32 (5) ◽  
pp. 1661-1672 ◽  
Author(s):  
DOMINIK KWIETNIAK ◽  
PIOTR OPROCHA
Keyword(s):  
Topological Dynamical System ◽  
Multiple Recurrence ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Open Questions ◽  
Weakly Mixing ◽  
Recurrence Theorem ◽  
Topological Weak Mixing ◽  
Mixing Property ◽  
Transitive System

AbstractThis article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:X→X is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is Δ-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

scholarly journals Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

2016 ◽  
Vol 37 (4) ◽  
pp. 1211-1237 ◽  
Author(s):  
FELIPE GARCÍA-RAMOS
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Haar Measure ◽  
Topological Category ◽  
Sequence Entropy ◽  
Ergodic Measures ◽  
Factor Map ◽  
Topological Dynamical Systems ◽  
Theoretical Sensitivity ◽  
Topological Sequence Entropy

We define weaker forms of topological and measure-theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum. We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to$\unicode[STIX]{x1D707}$-mean equicontinuity. In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is one–one on a set of full Haar measure). For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence, we obtain a dichotomy between discrete spectrum and a strong form of measure-theoretical sensitivity.

scholarly journals Topological sequence entropy and topologically weak mixing

2002 ◽  
Vol 66 (2) ◽  
pp. 207-211
Author(s):  
Simin Li
Keyword(s):  
Weak Mixing ◽  
Sequence Entropy ◽  
Topological Sequence Entropy

A charactersiation of topologically weak mixing is given by using the topological sequence entropy.

scholarly journals Weak mixing for locally compact quantum groups

2016 ◽  
Vol 37 (5) ◽  
pp. 1657-1680 ◽  
Author(s):  
AMI VISELTER
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebras ◽  
Unitary Representations ◽  
Discrete Groups ◽  
Weak Mixing ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Weakly Mixing

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

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