scholarly journals Uniform Convergence of Cesaro Averages for Uniquely Ergodic $C^*$-Dynamical Systems

2018 ◽  
Author(s):  
Francesco Fidaleo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Uniform Convergence ◽  
Unit Circle ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
C Algebra ◽  
Cesaro Averages ◽  
Uniformly Convergent ◽  
Uniquely Ergodic

Consider a uniquely ergodic $C^*$-dynamical system ba\-sed on a unital $*$-endomorphism $\Phi$ of a $C^*$-algebra. We prove the uniform convergence of Cesaro averages $\frac1{n}\sum_{k=0}^{n-1}\lambda^{-n}\Phi(a)$ for all values $\lambda$ in the unit circle which are not eigenvalues corresponding to "measurable non continuous" eigenfunctions. This result generalises the analogous one in commutative ergodic theory presented in [19], which turns out to be a combination of the Wiener-Wintner Theorem (cf. [22]) and the uniformly convergent ergodic theorem of Krylov and Bogolioubov (cf. [15]).

scholarly journals Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 987 ◽  
Author(s):  
Francesco Fidaleo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Uniform Convergence ◽  
Unit Circle ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
C Algebra ◽  
Cesaro Averages ◽  
Uniformly Convergent ◽  
Uniquely Ergodic

Consider a uniquely ergodic C * -dynamical system based on a unital *-endomorphism Φ of a C * -algebra. We prove the uniform convergence of Cesaro averages 1 n ∑ k = 0 n − 1 λ − n Φ ( a ) for all values λ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.

scholarly journals Existence of random invariant periodic curves via random semiuniform ergodic theorem

2016 ◽  
Vol 17 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Kenneth Uda
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Dissipative Structure ◽  
Random Dynamical Systems ◽  
Random Dynamical System ◽  
Compact Set

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system on a cylinder [Formula: see text] has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

Induced transformations of recurrent a.m.s. dynamical systems

2015 ◽  
Vol 15 (02) ◽  
pp. 1550010
Author(s):  
Sheng Huang ◽  
Mikael Skoglund
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Random Process ◽  
Ergodic Theorem ◽  
Finite Measure ◽  
Stationary Random Process ◽  
Finite State ◽  
Ratio Ergodic Theorem

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

scholarly journals Bounded complexity, mean equicontinuity and discrete spectrum

2019 ◽  
Vol 41 (2) ◽  
pp. 494-533 ◽  
Author(s):  
WEN HUANG ◽  
JIAN LI ◽  
JEAN-PAUL THOUVENOT ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Invariant Measure ◽  
Discrete Spectrum ◽  
Ergodic Theory ◽  
Topological Dynamics ◽  
Topological Dynamical System ◽  
The Mean ◽  
Bounded Complexity

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

scholarly journals Ideal structure of crossed products by endomorphisms via reversible extensions of C*-dynamical systems

2015 ◽  
Vol 26 (03) ◽  
pp. 1550022 ◽  
Author(s):  
Bartosz Kosma Kwaśniewski
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Transfer Operator ◽  
Crossed Product ◽  
Crossed Products ◽  
Ideal Structure ◽  
Gauge Invariant ◽  
C Algebra ◽  
System A ◽  
The Ideal

We consider an extendible endomorphism α of a C*-algebra A. We associate to it a canonical C*-dynamical system (B, β) that extends (A, α) and is "reversible" in the sense that the endomorphism β admits a unique regular transfer operator β⁎. The theory for (B, β) is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for (B, β) in terms of the initial system (A, α). We apply this idea to study the ideal structure of a non-unital version of the crossed product C*(A, α, J) introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal J in (ker α)⊥, and if J = ( ker α)⊥ it is a modification of Stacey's crossed product that works well with non-injective α's. We provide descriptions of the lattices of ideals in C*(A, α, J) consisting of gauge-invariant ideals and ideals generated by their intersection with A. We investigate conditions under which these lattices coincide with the set of all ideals in C*(A, α, J). In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of α or pointwise quasinilpotence of α.

scholarly journals Ergodic properties of the Anzai skew-product for the non-commutative torus

2020 ◽  
pp. 1-22 ◽  
Author(s):  
SIMONE DEL VECCHIO ◽  
FRANCESCO FIDALEO ◽  
LUCA GIORGETTI ◽  
STEFANO ROSSI
Keyword(s):  
Dynamical Systems ◽  
Unit Circle ◽  
Systematic Study ◽  
Cartesian Product ◽  
Skew Product ◽  
Quantum Dynamical ◽  
Ergodic Properties ◽  
Pointwise Limit ◽  
Quantum Dynamical Systems ◽  
Uniquely Ergodic

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $\unicode[STIX]{x1D6F7}$ on the non-commutative $2$ -torus $\mathbb{A}_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ , we investigate the pointwise limit, $\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$ , for $x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D706}$ a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.

scholarly journals On the Large Deviations Theorem of Weaker Types

2017 ◽  
Vol 27 (08) ◽  
pp. 1750127 ◽  
Author(s):  
Xinxing Wu ◽  
Xiong Wang ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Large Deviations ◽  
Ergodic Measure ◽  
Type I ◽  
Type Ii ◽  
Text Type ◽  
Type Iii ◽  
Ergodic Dynamical System ◽  
Uniquely Ergodic

In this paper, we introduce the concepts of the large deviations theorem of weaker types, i.e. type I, type I[Formula: see text], type II, type II[Formula: see text], type III, and type III[Formula: see text], and present a systematic study of the ergodic and chaotic properties of dynamical systems satisfying the large deviations theorem of various types. Some characteristics of the ergodic measure are obtained and then applied to prove that every dynamical system satisfying the large deviations theorem of type I[Formula: see text] is ergodic, which is equivalent to the large deviations theorem of type II[Formula: see text] in this regard, and that every uniquely ergodic dynamical system restricted on its support satisfies the large deviations theorem. Moreover, we prove that every dynamical system satisfying the large deviations theorem of type III is an [Formula: see text]-system.

MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

2011 ◽  
Vol 22 (01) ◽  
pp. 1-23 ◽  
Author(s):  
KAREN R. STRUNG ◽  
WILHELM WINTER
Keyword(s):  
Dynamical Systems ◽  
Compact Metric Space ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Minimal Dynamical Systems ◽  
Uniquely Ergodic ◽  
Stable Classification

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊α ℤ, and [Formula: see text] a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in [Formula: see text]. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra [Formula: see text], we prove that (C(X) ⋊α ℤ) ⊗ Mq∞ is tracially approximately [Formula: see text] if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞ is tracially approximately [Formula: see text]. If the class [Formula: see text] consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with [Formula: see text] for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊α ℤ or C*(C(X), uC0(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

scholarly journals On strictly weak mixing C *-dynamical systems and a weighted ergodic theorem

2010 ◽  
Vol 47 (2) ◽  
pp. 155-174
Author(s):  
Farrukh Mukhamedov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Tensor Product ◽  
Ergodic Theorem ◽  
Unique Ergodicity ◽  
Weak Mixing ◽  
Weighted Ergodic Theorem

We prove that unique ergodicity of tensor product of a C *-dynamical system implies its strictly weak mixing. By means of this result a uniform weighted ergodic theorem with respect to S -Besicovitch sequences for strictly weak mixing dynamical systems is proved. Moreover, we provide certain examples of strictly weak mixing dynamical systems.

Dynamical system approaches to climate variability

2020 ◽  
pp. 96-182
Author(s):  
Henk A. Dijkstra
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Climate Variability ◽  
Systems Theory ◽  
Ergodic Theory ◽  
Climate Dynamics ◽  
Dynamical Systems Analysis ◽  
Deterministic Dynamical Systems ◽  
Low Dimensional ◽  
System Approaches

A tutorial is provided on the application of dynamical systems theory to problems in climate dynamics. We start with the analysis of low-dimensional deterministic dynamical systems using bifurcation theory and provide examples in conceptual climate models.We then proceed to stochastic low-dimensional systems and eventually end with operator-based techniques within ergodic theory. In these notes, we start each section from a climate dynamics problem, motivate the choice of the model to study it, and use dynamical systems analysis to understand the behavior of the model solutions. In each of the chapters, a different phenomenon, a different type of model, and/or a different dynamical system tool will be presented.

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