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 Category Theory for Autonomous and Networked Dynamical Systems

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 302 ◽  
Author(s):  
Jean-Charles Delvenne
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Ergodic Theory ◽  
Category Theory ◽  
Open Systems ◽  
Topological Dynamics ◽  
Control Problems ◽  
Natural Conditions ◽  
Discussion Paper ◽  
Open Systems Theory

In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way.

scholarly journals Discrete spectrum for amenable group actions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tao Yu ◽  
Guohua Zhang ◽  
Ruifeng Zhang
Keyword(s):  
Invariant Measure ◽  
Discrete Spectrum ◽  
Invariant Measures ◽  
Hamming Distance ◽  
Amenable Group ◽  
Group Actions ◽  
The Mean

<p style='text-indent:20px;'>In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions.</p><p style='text-indent:20px;'>We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.</p>

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.

Sequence Entropy and Mild Mixing

1992 ◽  
Vol 44 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Qing Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Infinite Subset ◽  
Mixing Properties ◽  
Sequence Entropy ◽  
Number Of Authors

Entropy characterizations of different spectral and mixing properties of dynamical systems were dealt with by a number of authors (see [5], [6] and [8]).Given an infinite subset Γ = {tn}of N and a dynamical system (X, β,μ, T) one can define sequence entropy along for any finite Petition ξ, and hΓ(T) —supξ hΓ(T,ξ). In [6] Kushnirenko used sequence entropy to give a characterization of systems with discrete spectrum.

A variation on the variational principle and applications to entropy pairs

1997 ◽  
Vol 17 (1) ◽  
pp. 29-43 ◽  
Author(s):  
F. BLANCHARD ◽  
E. GLASNER ◽  
B. HOST
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Invariant Measure ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Open Cover ◽  
Topological Dynamical System ◽  
Positive Topological Entropy ◽  
Finite Open Cover

The variational principle states that the topological entropy of a topological dynamical system is equal to the sup of the entropies of invariant measures. It is proved that for any finite open cover there is an invariant measure such that the topological entropy of this cover is less than or equal to the entropies of all finer partitions. One consequence of this result is that for any dynamical system with positive topological entropy there exists an invariant measure whose set of entropy pairs is equal to the set of topological entropy pairs.

scholarly journals Differentiability with respect to parameters of average values in probabilistic contracting dynamical systems

1990 ◽  
Vol 10 (3) ◽  
pp. 599-610 ◽  
Author(s):  
W. Douglas Withers
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Harmonic Function ◽  
Invariant Measure ◽  
Compact Subset ◽  
Julia Set ◽  
Spatial Variable ◽  
External Parameter ◽  
Average Value ◽  
Complex Functions

AbstractWe consider a dynamical system consisting of a compact subset of RN or CN with several contracting maps chosen with prescribed probabilities, which may depend on position. We show that if the maps and the probabilities are Cl+α functions of the spatial variable and an external parameter, then the average value of a Cl+α function is a differentiate function of the parameter. One implication of this theorem is that for certain families of complex functions dependent on a parameter the reciprocal of the dimension of an invariant measure on the Julia set is a harmonic function of the parameter.

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].

Classical and quantum ergodicity

2017 ◽  
Author(s):  
Christopher D. Sogge
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Ergodic Theory ◽  
High Frequency ◽  
Limit Theorems ◽  
Initial Development ◽  
Quantum Ergodicity ◽  
Physical Problems ◽  
Pure Mathematics ◽  
Important Branch

This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the end of the nineteenth century. The word ergodic has as its roots two Greek words: ergon, meaning work or energy, and hodos, meaning path or way. Even though ergodic theory's initial development was motivated by physical problems, it has become an important branch of pure mathematics that studies dynamical systems possessing an invariant measure. Thus, this chapter first presents some of the basic limit theorems that are key to the classical theory. It then turns to quantum ergodicity.

Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1

1993 ◽  
Vol 13 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Nobuo Aoki ◽  
Jun Tomiyama
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Transformation Group ◽  
Irreducible Representations ◽  
Topological Dynamical System ◽  
Compact Metric Space ◽  
C Algebras ◽  
Topological Dynamical Systems

AbstractFor a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

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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