scholarly journals Characteristic measures of symbolic dynamical systems

2021 ◽  
pp. 1-7
Author(s):  
JOSHUA FRISCH ◽  
OMER TAMUZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Automorphism Group ◽  
Probability Measure ◽  
Automorphism Groups ◽  
Topological Dynamical System ◽  
Zero Entropy ◽  
Characteristic Measure

Abstract A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

A minimal distal map on the torus with sub-exponential measure complexity

2018 ◽  
Vol 40 (4) ◽  
pp. 953-974 ◽  
Author(s):  
WEN HUANG ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Borel Probability Measure ◽  
Skew Product ◽  
Topological Dynamical System ◽  
Borel Probability ◽  
Product Map

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

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.

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.

scholarly journals Zero-dimensional isomorphic dynamical models

2018 ◽  
Vol 40 (8) ◽  
pp. 2116-2130
Author(s):  
TOMASZ DOWNAROWICZ ◽  
LEI JIN ◽  
WOLFGANG LUSKY ◽  
YIXIAO QIAO
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Extreme Points ◽  
Dimensional System ◽  
Choquet Simplex ◽  
Countable Union ◽  
Topological Dynamical System ◽  
Dynamical Models ◽  
Bauer Simplex

By an assignment we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems obeying some natural restrictions. We prove that if $\unicode[STIX]{x1D6F7}$ is an aperiodic assignment on a Choquet simplex $K$ such that the set of extreme points $\mathsf{ex}K$ is a countable union $\bigcup _{n}E_{n}$, where each set $E_{n}$ is compact, zero-dimensional and the restriction of $\unicode[STIX]{x1D6F7}$ to the Bauer simplex $K_{n}$ spanned by $E_{n}$ can be ‘embedded’ in some topological dynamical system, then $\unicode[STIX]{x1D6F7}$ can be ‘realized’ in a zero-dimensional system.

scholarly journals The structure of tame minimal dynamical systems

2007 ◽  
Vol 27 (6) ◽  
pp. 1819-1837 ◽  
Author(s):  
ELI GLASNER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Probability Measure ◽  
Topological Space ◽  
Haar Measure ◽  
Invariant Probability Measure ◽  
Structure Theory ◽  
Convergence Of Sequences ◽  
Minimal Dynamical Systems ◽  
Minimal Dynamical System

AbstractA dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \mathbb {N}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

scholarly journals On forward iterated Hausdorffness and development of embryo from zygote in bitopological dynamical systems

2021 ◽  
Vol 13(62) (2) ◽  
pp. 399-410
Author(s):  
Santanu Acharjee ◽  
Kabindra Goswami ◽  
Hemanta Kumar Sarmah
Keyword(s):  
Dynamical System ◽  
Cell Division ◽  
Dynamical Systems ◽  
Topological Space ◽  
Development Process ◽  
Hausdorff Space ◽  
Topological Dynamical System ◽  
Relative Topology ◽  
Dynamical Properties

Topological dynamical system is an area of dynamical system to investigate dynamical properties in terms of a topological space. Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] applied topological dynamical system to explore the development process of an embryo from the zygote until birth and made three conjectures. In this paper, we disprove conjecture 3 of Nada and Zohny [Nada, S.I. and Zohny, H., An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202-204] by applying some of our mathematical results of bitopological dynamical system. Also, we introduce forward iterated Hausdorff space, backward iterated Hausdorff space, pairwise iterated Hausdor_ space and establish relations between them in bitopological dynamical system. We formulate the function that represents cell division (specially, mitosis) and using this function we show that in the development process of a human baby from the zygote until its birth, there is a stage where the developing stage is forward iterated Hausdorff

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

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