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

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.

Topological stability and shadowing of dynamical systems from measure theoretical viewpoint

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.

The G -Asymptotic Tracking Property and G -Asymptotic Average Tracking Property in the Inverse Limit Spaces under Group Action

Firstly, we introduce the definitions of G -asymptotic tracking property, G -asymptotic average tracking property, and G -quasi-weak almost-periodic point. Secondly, we study their dynamical properties and characteristics. The results obtained improve the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space and provide the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, and computer science.

Modulated crystals and almost periodic measures

Modulated 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.

Dispersive semiflows on fiber bundles

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.

ON THE STRENGTH OF TWO RECURRENCE THEOREMS

This 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.

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

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

Minimal flows of finite almost periodic rank

The 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.

Recurrent and weakly recurrent points inβG

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.