Topological Ergodic Shadowing and Chaos on Uniform Spaces

2018 ◽  
Vol 28 (03) ◽  
pp. 1850043 ◽  
Author(s):  
Xinxing Wu ◽  
Xin Ma ◽  
Zhu Zhu ◽  
Tianxiu Lu
Keyword(s):  
Dynamical System ◽  
Uniform Space ◽  
Uniform Spaces ◽  
Weakly Mixing ◽  
Topologically Mixing

This paper firstly proves that every dynamical system defined on a Hausdorff uniform space with topologically ergodic shadowing is topologically mixing, thus topologically chain mixing. Then, the following is proved: (1) every weakly mixing dynamical system defined on a second countable Baire–Hausdorff uniform space is chaotic in the sense of both Li–Yorke and Auslander–Yorke; (2) every point transitive dynamical system defined on a Hausdorff uniform space is either almost equicontinuous or sensitive.

Invariants of Uniform Conjugacy on Uniform Dynamical System

2021 ◽  
Vol 26 (4) ◽  
Author(s):  
Alaa Saeed Abboud ◽  
Ihsan Jabbar Khadim
Keyword(s):  
Dynamical System ◽  
Uniform Space ◽  
Uniform Spaces

  In this paper, we present some important dynamical concepts on uniform space such as the uniform minimal systems, uniform shadowing, and strong uniform shadowing. We explain some definitions and theorems such as definition uniform expansive, weak uniform expansive, uniform generator, and the proof of the theorems for them. We prove that if  be a homeomorphism on a compact uniform space  then  has uniform shadowing if and only if has uniform shadowing, so if  has strong uniform shadowing if and only if has strong uniform shadowing. We also show that  and  be two uniform homeomorphisms on compact uniform spaces  and , if  is a uniform conjugacy from  to , then . Besides some other results.

Transitivity, weakly mixing property and chaos

2016 ◽  
Vol 30 (02) ◽  
pp. 1550274 ◽  
Author(s):  
Lidong Wang ◽  
Jianhua Liang ◽  
Yiyi Wang ◽  
Xuelian Sun
Keyword(s):  
Dynamical System ◽  
Periodic Point ◽  
Metric Space ◽  
Strong Sense ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing ◽  
Topologically Mixing ◽  
Mixing Property

Let [Formula: see text] be a compact metric space without isolated points and let [Formula: see text] be a continuous map. In this paper, if [Formula: see text] is a transitive dynamical system with a repelling periodic point, then [Formula: see text] is chaotic in the sense of Kato. In addition, if [Formula: see text] is weakly topologically mixing, then [Formula: see text] is chaotic in the strong sense of Kato.

The Mean Sensitivity and Mean Equicontinuity in Uniform Spaces

2020 ◽  
Vol 30 (08) ◽  
pp. 2050122 ◽  
Author(s):  
Xinxing Wu ◽  
Shudi Liang ◽  
Xin Ma ◽  
Tianxiu Lu ◽  
Seyyed Alireza Ahmadi
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Inverse Limit ◽  
Uniform Space ◽  
Uniform Spaces ◽  
The Mean ◽  
Mean Sensitivity

Some characteristics of mean sensitivity and Banach mean sensitivity using Furstenberg families and inverse limit dynamical systems are obtained. The iterated invariance of mean sensitivity and Banach mean sensitivity are proved. Applying these results, the notion of mean sensitivity and Banach mean sensitivity is extended to uniform spaces. It is proved that a point-transitive dynamical system in a Hausdorff uniform space is either almost (Banach) mean equicontinuous or (Banach) mean sensitive.

On ARU-Resolutions of Uniform Spaces

2003 ◽  
Vol 10 (2) ◽  
pp. 201-207
Author(s):  
V. Baladze
Keyword(s):  
Uniform Space ◽  
Uniform Spaces

Abstract In this paper theorems which give conditions for a uniform space to have an ARU-resolution are proved. In particular, a finitistic uniform space admits an ARU-resolution if and only if it has trivial uniform shape or it is an absolute uniform shape retract.

On Homology and Cohomology Groups of Remainders

2004 ◽  
Vol 11 (4) ◽  
pp. 613-633
Author(s):  
V. Baladze ◽  
L. Turmanidze
Keyword(s):  
Uniform Space ◽  
Cohomology Groups ◽  
Uniform Spaces ◽  
Intrinsic Characterization

Abstract Border homology and cohomology groups of pairs of uniform spaces are defined and studied. These groups give an intrinsic characterization of Čech type homology and cohomology groups of the remainder of a uniform space.

scholarly journals A study of uniformities on the space of uniformly continuous mappings

2020 ◽  
Vol 18 (1) ◽  
pp. 1478-1490
Author(s):  
Ankit Gupta ◽  
Abdulkareem Saleh Hamarsheh ◽  
Ratna Dev Sarma ◽  
Reny George
Keyword(s):  
Uniform Space ◽  
Uniform Spaces ◽  
Continuous Mappings ◽  
Uniformly Continuous

Abstract New families of uniformities are introduced on UC(X,Y) , the class of uniformly continuous mappings between X and Y, where (X,{\mathcal{U}}) and (Y,{\mathcal{V}}) are uniform spaces. Admissibility and splittingness are introduced and investigated for such uniformities. Net theory is developed to provide characterizations of admissibility and splittingness of these spaces. It is shown that the point-entourage uniform space is splitting while the entourage-entourage uniform space is admissible.

scholarly journals On expansive homeomorphism of uniform spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 292-304
Author(s):  
Ali Barzanouni ◽  
Ekta Shah
Keyword(s):  
Uniform Space ◽  
Regular Space ◽  
Uniform Spaces ◽  
Expansive Homeomorphism

Abstract We study the notion of expansive homeomorphisms on uniform spaces. It is shown that if there exists a topologically expansive homeomorphism on a uniform space, then the space is always a Hausdor space and hence a regular space. Further, we characterize orbit expansive homeomorphisms in terms of topologically expansive homeomorphisms and conclude that if there exist a topologically expansive homeomorphism on a compact uniform space then the space is always metrizable.

scholarly journals Compactifications of totally bounded quasi-uniform spaces

1986 ◽  
Vol 28 (1) ◽  
pp. 31-36 ◽  
Author(s):  
P. Fletcher ◽  
W. F. Lindgren
Keyword(s):  
Hausdorff Space ◽  
Uniform Space ◽  
Continuous Extension ◽  
Sufficient Condition ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Totally Bounded ◽  
Continuous Map ◽  
Total Boundedness ◽  
Uniformly Continuous

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

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