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 Neutrosophic Orbit Topological Spaces

2021 ◽  
Vol 18 (24) ◽  
pp. 1443
Author(s):  
T Madhumathi ◽  
F NirmalaIrudayam
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Space ◽  
Necessary Conditions ◽  
Neutrosophic Set ◽  
Topological Spaces ◽  
Open Set ◽  
Evolution Function

Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. In the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. Hence in this paper we focus on introducing the concept of neutrosophic orbit topological space denoted as (X, tNO). Also, some of the important characteristics of neutrosophic orbit open sets are discussed with suitable examples. HIGHLIGHTS The orbit in mathematics has an important role in the study of dynamical systems Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. We combine the above two topics and create the following new concept The collection of all neutrosophic orbit open sets under the mapping . we introduce the necessary conditions on the mapping 𝒇 in order to obtain a fixed orbit of a neutrosophic set (i.e., 𝒇(𝝁) = 𝝁) for any neutrosophic orbit open set 𝝁 under the mapping 𝒇

scholarly journals Haar measure and compact right topological groups

1992 ◽  
Vol 45 (3) ◽  
pp. 399-413 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Topological Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Structure Theorem ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Positive Answer ◽  
Topological Groups ◽  
Low Dimensional ◽  
Compact Right Topological Group

The consideration of compact right topological groups goes back at least to a paper of Ellis in 1958, where it is shown that a flow is distal if and only if the enveloping semigroup of the flow is such a group (now called the Ellis group of the distal flow). Later Ellis, and also Namioka, proved that a compact right topological group admits a left invariant probability measure. As well, Namioka proved that there is a strong structure theorem for compact right topological groups. More recently, John Pym and the author strengthened this structure theorem enough to be able to establish the existence of Haar measure on a compact right topological group, a probability measure that is invariant under all continuous left and right translations, and is unique as such. Examples of compact right topological groups have been considered earlier. In the present paper, we give concrete representations of several Ellis groups coming from low dimensional nilpotent Lie groups. We study these compact right topological groups, and two others, in some detail, paying attention in particular to the structure theorem and Haar measure, and to the question: is Haar measure uniquely determined by left invariance alone? (It is uniquely determined by right invariance alone.) To assist in answering this question, we develop some sufficient conditions for a positive answer. We suspect that one of the examples, a compact right topological group coming from the Euclidean group of the plane, does not satisfy these conditions; we don't know if the question has a positive answer for this group.

Algorithmic complexity of points in dynamical systems

1993 ◽  
Vol 13 (4) ◽  
pp. 807-830 ◽  
Author(s):  
Homer S. White
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Invariant Probability Measure ◽  
Algorithmic Complexity ◽  
Maximal Entropy ◽  
Property A ◽  
Open Set ◽  
Unique Measure ◽  
Set Of Points

AbstractThis work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.

scholarly journals Convergence in distribution of the multi-dimensional Kohonen algorithm

2001 ◽  
Vol 38 (1) ◽  
pp. 136-151 ◽  
Author(s):  
Ali A. Sadeghi
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Topological Space ◽  
Wide Class ◽  
Invariant Probability Measure ◽  
Learning Rate ◽  
Important Case ◽  
Convergence In Distribution ◽  
Special Shape ◽  
Data Space

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

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.

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

scholarly journals Convergence in distribution of the multi-dimensional Kohonen algorithm

2001 ◽  
Vol 38 (01) ◽  
pp. 136-151
Author(s):  
Ali A. Sadeghi
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Topological Space ◽  
Wide Class ◽  
Invariant Probability Measure ◽  
Learning Rate ◽  
Important Case ◽  
Convergence In Distribution ◽  
Special Shape ◽  
Data Space

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

scholarly journals Effectivity of uniqueness of the maximal entropy measure on -adic homogeneous spaces

2015 ◽  
Vol 36 (6) ◽  
pp. 1972-1988 ◽  
Author(s):  
RENE RÜHR
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Algebraic Group ◽  
Homogeneous Spaces ◽  
Regular Representation ◽  
Invariant Probability Measure ◽  
Linear Algebraic Group ◽  
Maximal Entropy ◽  
Entropy Measure ◽  
Effective Version

We consider the dynamical system given by an $\text{Ad}$-diagonalizable element $a$ of the $\mathbb{Q}_{p}$-points $G$ of a unimodular linear algebraic group acting by translation on a finite volume quotient $X$. Assuming that this action is exponentially mixing (e.g. if $G$ is simple) we give an effective version (in terms of $K$-finite vectors of the regular representation) of the following statement: If ${\it\mu}$ is an $a$-invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$, then ${\it\mu}$ is close to the unique $G$-invariant probability measure of $X$.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

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.

Sign in / Sign up

Export Citation Format

Share Document