scholarly journals Topological Multiple Recurrence of Weakly Mixing Minimal Systems for Generalized Polynomials

2021 ◽  
Vol 37 (12) ◽  
pp. 1847-1874
Author(s):  
Rui Feng Zhang ◽  
Jian Jie Zhao
Keyword(s):  
Multiple Recurrence ◽  
Generalized Polynomials ◽  
Weakly Mixing

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 A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

Mixing for stationary processes with finite-order multiple Wiener-Itô integral representation

1996 ◽  
Vol 16 (5) ◽  
pp. 1087-1100
Author(s):  
Eric Slud ◽  
Daniel Chambers
Keyword(s):  
Large Class ◽  
Order Term ◽  
Sufficient Conditions ◽  
Polynomial Form ◽  
Itô Integral ◽  
Ito Integral ◽  
Weakly Mixing ◽  
Subordinated Processes ◽  
Infinite Sums ◽  
Necessary And Sufficient

abstractNecessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

Exponential operators, generalized polynomials and evolution problems

2001 ◽  
Vol 61 (2) ◽  
pp. 99-108 ◽  
Author(s):  
G Dattoli ◽  
A.M Mancho ◽  
M Quattromini ◽  
A Torre
Keyword(s):  
Generalized Polynomials ◽  
Evolution Problems

scholarly journals A NONCONVENTIONAL STRONG LAW OF LARGE NUMBERS AND FRACTAL DIMENSIONS OF SOME MULTIPLE RECURRENCE SETS

2012 ◽  
Vol 12 (03) ◽  
pp. 1150023 ◽  
Author(s):  
YURI KIFER
Keyword(s):  
Law Of Large Numbers ◽  
Fractal Dimensions ◽  
Growth Conditions ◽  
Multiple Recurrence ◽  
Strong Law ◽  
Moment Conditions ◽  
Vector Process ◽  
Regularity Properties ◽  
Large Numbers ◽  
Positive Functions

We provide conditions which yield a strong law of large numbers for expressions of the form [Formula: see text] where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polinomial growth and certain regularity properties and qi, i > m are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places q1(n), …, qℓ(n).

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