scholarly journals Topological stability and shadowing of dynamical systems from measure theoretical viewpoint

2021 ◽  
pp. 1-15
Author(s):  
Jiandong Yin ◽  
Meihua Dong
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Periodic Point ◽  
Almost Periodic ◽  
Topological Stability ◽  
Shadowing Property ◽  
Continuous Map ◽  
Support Measure ◽  
Stable Invariant ◽  
Almost Periodic Point

In this paper it is proved that a topologically stable invariant measure has no sinks or sources in its support; an expansive homeomorphism is topologically stable if it exhibits a topologically stable nonatomic Borel support measure and a continuous map has the shadowing property if there exists an invariant measure with the shadowing property such that each almost periodic point is contained in the support of the invariant measure.

scholarly journals G-Expansibility and G-Almost Periodic Point under Topological Group Action

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Zhanjiang Ji
Keyword(s):  
Topological Group ◽  
Periodic Point ◽  
Inverse Limit ◽  
The Self ◽  
Almost Periodic ◽  
Shadowing Property ◽  
Limit Shadowing ◽  
Shift Map ◽  
Inverse Limit Space ◽  
Almost Periodic Point

Firstly, the new concepts of G − expansibility, G − almost periodic point, and G − limit shadowing property were introduced according to the concepts of expansibility, almost periodic point, and limit shadowing property in this paper. Secondly, we studied their dynamical relationship between the self-map f and the shift map σ in the inverse limit space under topological group action. The following new results are obtained. Let X , d be a metric G − space and X f , G ¯ ,   d ¯ , σ be the inverse limit space of X , G , d , f . (1) If the map f : X ⟶ X is an equivalent map, then we have A P G ¯ σ = Lim ← A p G f , f . (2) If the map f : X ⟶ X is an equivalent surjection, then the self-map f is G − expansive if and only if the shift map σ is G ¯ − expansive. (3) If the map f : X ⟶ X is an equivalent surjection, then the self-map f has G − limit shadowing property if and only if the shift map σ has G ¯ − limit shadowing property. The conclusions of this paper generalize the corresponding results given in the study by Li, Niu, and Liang and Li . Most importantly, it provided the theoretical basis and scientific foundation for the application of tracking property in computational mathematics and biological mathematics.

Dynamics of G-processes

2020 ◽  
Vol 20 (01) ◽  
pp. 2050037
Author(s):  
W. Jung ◽  
K. Lee ◽  
C.A. Morales
Keyword(s):  
Dynamical Systems ◽  
Group Structure ◽  
Global Solutions ◽  
Topological Stability ◽  
Natural Group ◽  
Shadowing Property ◽  
Infinite Dimensional ◽  
Mathematical Sciences ◽  
The Relationship

A G-process is briefly a process ([A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182 (Springer, 2013)], [C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971) 239–252], [P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176 (Amer. Math. Soc., 2011)]) for which the role of evolution parameter is played by a general topological group [Formula: see text]. These processes are broad enough to include the [Formula: see text]-actions (characterized as autonomous [Formula: see text]-processes) and the two-parameter flows (where [Formula: see text]). We endow the space of [Formula: see text]-processes with a natural group structure. We introduce the notions of orbit, pseudo-orbit and shadowing property for [Formula: see text]-processes and analyze the relationship with the [Formula: see text]-processes group structure. We study the equicontinuous [Formula: see text]-processes and use it to construct nonautonomous [Formula: see text]-processes with the shadowing property. We study the global solutions of the [Formula: see text]-processes and the corresponding global shadowing property. We study the expansivity (global and pullback) of the [Formula: see text]-processes. We prove that there are nonautonomous expansive [Formula: see text]-processes and characterize the existence of expansive equicontinuous [Formula: see text]-processes. We define the topological stability for [Formula: see text]-processes and prove that every expansive [Formula: see text]-process with the shadowing property is topologically stable. Examples of nonautonomous topologically stable [Formula: see text]-processes are given.

scholarly journals Dispersive semiflows on fiber bundles

2017 ◽  
Vol 37 (2) ◽  
pp. 85-99
Author(s):  
Josiney A. Souza ◽  
Hélio V. M. Tozatti
Keyword(s):  
Periodic Point ◽  
Fiber Bundle ◽  
Vector Bundles ◽  
Base Space ◽  
Total Space ◽  
Fiber Bundles ◽  
Almost Periodic ◽  
Special Result ◽  
Almost Periodic Point

This paper studies dispersiveness of semiflows on fiber bundles. The main result says that a right invariant semiflow on a fiber bundle is dispersive on the base space if and only if there is no almost periodic point and the semiflow is dispersive on the total space. A special result states that linear semiflows on vector bundles are not dispersive.

scholarly journals Minimal flows of finite almost periodic rank

1988 ◽  
Vol 8 (2) ◽  
pp. 155-172 ◽  
Author(s):  
Joseph Auslander ◽  
Nelson Markley
Keyword(s):  
Positive Integer ◽  
Periodic Point ◽  
Power Factor ◽  
Finite Rank ◽  
Minimal Flow ◽  
Almost Periodic ◽  
Product Flow ◽  
Structural Differences ◽  
Proximal Extension ◽  
Almost Periodic Point

AbstractThe totally minimal flow (X, T) is said to have finite almost periodic rank if there is a positive integer n such that whenever (x1, x2,…, xn+1) is an almost periodic point of the product flow (Xn+1, T×…×T) then, for some i≠j, xi, and xj are in the same orbit. The rank of (X, T) is the smallest such integer. If (Y, S) is a graphic flow, (Y, Sn) has rank |n| and it is shown that every finite rank flow has, modulo a proximal extension, a graphic power factor. Various classes of finite rank flows are defined, and characterized in terms of their Ellis groups. There are four disjoint types which have basic structural differences.

Weakly Almost Periodic Points and Some Chaotic Properties of Dynamical Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550115 ◽  
Author(s):  
Jiandong Yin ◽  
Zuoling Zhou
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Full Measure ◽  
Almost Periodic ◽  
Countable Base ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Weakly Almost Periodic

Let X be a compact metric space and f : X → X be a continuous map. In this paper, ergodic chaos and strongly ergodic chaos are introduced, and it is proven that f is strongly ergodically chaotic if f is transitive but not minimal and has a full measure center. In addition, some sufficient conditions for f to be Ruelle–Takens chaotic are presented. For instance, we prove that f is Ruelle–Takens chaotic if f is transitive and there exists a countable base [Formula: see text] of X such that for each i > 0, the meeting time set N(Ui, Ui) for Ui with respect to itself has lower density larger than [Formula: see text].

scholarly journals ON THE STRENGTH OF TWO RECURRENCE THEOREMS

2016 ◽  
Vol 81 (4) ◽  
pp. 1357-1374 ◽  
Author(s):  
ADAM R. DAY
Keyword(s):  
Periodic Point ◽  
Reverse Mathematics ◽  
Topological Dynamics ◽  
Almost Periodic ◽  
Image Position ◽  
Almost Periodic Point

AbstractThis paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between WKL and ACA (working over RCA0). This is the first example of a theorem with this property. It also shows the existence of an almost periodic point is conservative over RCA0for${\rm{\Pi }}_1^1$-sentences.

scholarly journals On some types of almost periodic point in Bi-Topological dynamcs

2005 ◽  
Vol 2 (1) ◽  
pp. 143-147
Author(s):  
Baghdad Science Journal
Keyword(s):  
Periodic Point ◽  
Periodic Points ◽  
Topological Dynamics ◽  
Almost Periodic ◽  
Almost Periodic Point

In this paper We introduce some new types of almost bi-periodic points in topological bitransfprmation groups and thier effects on some types of minimaliy in topological dynamics

scholarly journals Recurrent and weakly recurrent points inβG

1986 ◽  
Vol 9 (2) ◽  
pp. 319-322 ◽  
Author(s):  
Mostafa Nassar
Keyword(s):  
Periodic Point ◽  
Almost Periodic ◽  
Recurrent Point ◽  
Almost Periodic Point

It is shown in this paper that ifβGis the Stone-Čech compactification of a groupG, andGsatisfying a certain condition, then there is a weakly recurrent point inβGwhich is not almost periodic, and if another condition will be added, then there is a recurrent point inβGwhich is not almost periodic point.

scholarly journals Modulated crystals and almost periodic measures

2020 ◽  
Vol 110 (12) ◽  
pp. 3435-3472
Author(s):  
Jeong-Yup Lee ◽  
Daniel Lenz ◽  
Christoph Richard ◽  
Bernd Sing ◽  
Nicolae Strungaru
Keyword(s):  
Periodic Point ◽  
Diffraction Theory ◽  
Almost Periodic ◽  
Point Sets ◽  
De Bruijn ◽  
Almost Periodic Point

AbstractModulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyze these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similar to de Bruijn’s analysis, we find stability with respect to almost periodic modulations.

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