scholarly journals Null systems in the non-minimal case

2019 ◽  
Vol 40 (12) ◽  
pp. 3420-3437
Author(s):  
JIAHAO QIU ◽  
JIANJIE ZHAO
Keyword(s):  
Dynamical System ◽  
Direct Application ◽  
Minimal Points ◽  
Null System ◽  
Proximal Relation

In this paper, it is shown that if a dynamical system is null and distal, then it is equicontinuous. It turns out that a null system with closed proximal relation is mean equicontinuous. As a direct application, it follows that a null dynamical system with dense minimal points is also mean equicontinuous. Meanwhile, a distal system with trivial $\text{Ind}_{\text{fip}}$-pairs and a non-trivial regionally proximal relation of order $\infty$ are constructed.

Topological Sequence Entropy and Chaos

2014 ◽  
Vol 24 (07) ◽  
pp. 1450100
Author(s):  
Xin Liu ◽  
Huoyun Wang ◽  
Heman Fu
Keyword(s):  
Dynamical System ◽  
Sequence Entropy ◽  
Null System ◽  
Upper Density ◽  
Topological Sequence Entropy

A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Given 0 ≤ p ≤ q ≤ 1, a dynamical system is [Formula: see text] chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. It shows that, for any 0 ≤ p < q ≤ 1 or p = q = 0 or p = q = 1, a dynamical system which is null and [Formula: see text] chaotic can be realized.

scholarly journals On $ n $-tuplewise IP-sensitivity and thick sensitivity

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jian Li ◽  
Yini Yang
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Necessary Conditions ◽  
Minimal System ◽  
Topological Dynamical System ◽  
Minimal Points ◽  
Sufficient And Necessary Conditions ◽  
Weakly Mixing ◽  
Factor Maps

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> be a topological dynamical system and <inline-formula><tex-math id="M3">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>. We say that <inline-formula><tex-math id="M4">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive (resp. <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive) if there exists a constant <inline-formula><tex-math id="M7">\begin{document}$ \delta&gt;0 $\end{document}</tex-math></inline-formula> with the property that for each non-empty open subset <inline-formula><tex-math id="M8">\begin{document}$ U $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M9">\begin{document}$ X $\end{document}</tex-math></inline-formula>, there exist <inline-formula><tex-math id="M10">\begin{document}$ x_1,x_2,\dotsc,x_n\in U $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Bigl\{k\in \mathbb{N}\colon \min\limits_{1\le i&lt;j\le n}d(T^k x_i,T^k x_j)&gt;\delta\Bigr\} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an IP-set (resp. a thick set).</p><p style='text-indent:20px;'>We obtain several sufficient and necessary conditions of a dynamical system to be <inline-formula><tex-math id="M11">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive or <inline-formula><tex-math id="M12">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive for all <inline-formula><tex-math id="M14">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, while it is <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive if and only if it has at least <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula> minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP<inline-formula><tex-math id="M17">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP<inline-formula><tex-math id="M18">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP<inline-formula><tex-math id="M19">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.</p>

Contributions to the mechanisms of mitosis and roles of cytoplasmic microtubules from ultrastructural analyses of fungi

1978 ◽  
Vol 36 (2) ◽  
pp. 266-267
Author(s):  
I. Brent Heath
Keyword(s):  
Nuclear Envelope ◽  
Direct Application ◽  
Ultrastructural Analysis ◽  
General Interest ◽  
Research Results ◽  
Organelle Motility ◽  
Cytoplasmic Microtubules

Detailed ultrastructural analysis of fungal mitotic systems and cytoplasmic microtubules might be expected to contribute to a number of areas of general interest in addition to the direct application to the organisms of study. These areas include possibly fundamental general mechanisms of mitosis; evolution of mitosis; phylogeny of organisms; mechanisms of organelle motility and positioning; characterization of cellular aspects of microtubule properties and polymerization control features. This communication is intended to outline our current research results relating to selected parts of the above questions.Mitosis in the oomycetes Saprolegnia and Thraustotheca has been described previously. These papers described simple kinetochores and showed that the kineto- chores could probably be used as markers for the poorly defined chromosomes. Kineto- chore counts from serially sectioned prophase mitotic nuclei show that kinetochore replication precedes centriole replication to yield a single hemispherical array containing approximately the 4 n number of kinetochore microtubules diverging from the centriole associated "pocket" region of the nuclear envelope (Fig. 1).

Sign in / Sign up

Export Citation Format

Share Document