Multiple recurrence and almost sure convergence for weakly mixing dynamical systems

1998 ◽  
Vol 103 (1) ◽  
pp. 111-124 ◽  
Author(s):  
I. Assani
Keyword(s):  
Dynamical Systems ◽  
Almost Sure Convergence ◽  
Multiple Recurrence ◽  
Weakly Mixing

scholarly journals Partitions with independent iterates in random dynamical systems

2009 ◽  
Vol 30 (2) ◽  
pp. 361-377 ◽  
Author(s):  
BORIS BEGUN ◽  
ANDRÉS DEL JUNCO
Keyword(s):  
Dynamical Systems ◽  
Random Dynamical Systems ◽  
Weakly Mixing ◽  
Dense Set ◽  
Measure Preserving Actions

AbstractKrengel characterized weakly mixing actions (X,T) as those measure-preserving actions having a dense set of partitions of X with infinitely many jointly independent images under iterates of T. Using the tools developed in later papers—one by del Junco, Reinhold and Weiss, another by del Junco and Begun—we prove analogues of these results for weakly mixing random dynamical systems (in other words, relatively weakly mixing systems).

WEAKLY MIXING IMPLIES DISTRIBUTIONAL CHAOS IN A SEQUENCE

2010 ◽  
Vol 24 (14) ◽  
pp. 1595-1600 ◽  
Author(s):  
LIDONG WANG ◽  
YU'E YANG ◽  
ZHENYAN CHU ◽  
GONGFU LIAO
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Distributional Chaos ◽  
Weakly Mixing ◽  
Separable Metric Space

Let X be a separable metric space containing at least two points, and f:X→X be continuous. In this paper, we consider dynamical systems on X, and prove that topologically weakly mixing implies distributional chaos in a sequence.

scholarly journals Bergelson's theorem for weakly mixing C*-dynamical systems

2009 ◽  
Vol 192 (3) ◽  
pp. 235-257 ◽  
Author(s):  
Rocco Duvenhage
Keyword(s):  
Dynamical Systems ◽  
Weakly Mixing

scholarly journals Non-recurrence sets for weakly mixing linear dynamical systems

2012 ◽  
Vol 34 (1) ◽  
pp. 132-152 ◽  
Author(s):  
SOPHIE GRIVAUX
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Gaussian Measure ◽  
Bounded Linear Operator ◽  
Probability Space ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Linear Dynamical Systems ◽  
Weakly Mixing

AbstractWe study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space$X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets$\{n_{k}\}$such that$n_{k+1}/n_{k}\to +\infty $, or such that$n_{k}$divides$n_{k+1}$for each$k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each$r\ge 1$, of$r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

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 Some Chaotic Properties of Discrete Fuzzy Dynamical Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yaoyao Lan ◽  
Qingguo Li ◽  
Chunlai Mu ◽  
Hua Huang
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Continuous Map ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Fuzzy Dynamical Systems ◽  
Compact Subsets

Letting(X,d)be a metric space,f:X→Xa continuous map, and(ℱ(X),D)the space of nonempty fuzzy compact subsets ofXwith the Hausdorff metric, one may study the dynamical properties of the Zadeh's extensionf̂:ℱ(X)→ℱ(X):u↦f̂u. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations betweenfandf̂. More specifically, we study the transitivity, weakly mixing, periodic density in system(X,f), and its connections with the same ones in its fuzzified system.

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.

Sign in / Sign up

Export Citation Format

Share Document