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In this paper, the concept of a ternary relation (named as C-relation) is introduced. It is observed that every closure operator can be used to define a C-relation and conversely, any C-relation induces a closure operator. Thus, topological concepts can be studied in terms of relations. [ABSTRACT FROM AUTHOR]

In this manuscript, we aim to investigate the topological entropy defined on compact uniform spaces. First, we introduce the notions of measure-theoretic entropy and topological entropy of subsets on compact uniform spaces. Subsequently, we obtain a variational principle for topological entropy of subsets on compact uniform spaces. [ABSTRACT FROM AUTHOR]

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We introduce topological definitions of strong chain transitivity and we prove that topological average shadowing and topological pseudo-orbital specification each imply topological strong chain transitivity. Furthermore, we introduce the notion of the topological Lipschitz property and we show that it is a sufficient condition for the coincidence of the strong chain recurrent set of a dynamical system and its iterations. [ABSTRACT FROM AUTHOR]

This manuscript extends the nonautonomous notions of attractor-repeller pair and Morse decomposition from the setting of nonautonomous dynamical systems to the setting of topological cocycles. It takes into consideration a general notion of cocycle governed by a group action $ G\times P\rightarrow P $. The limit behavior is determined by the directional attractors depending on a filter basis on the subsets of a semigroup $ S\subset G $. The Conley methodology of constructing Morse decompositions is extended by means of directional attractor-repeller pairs. The main result shows the existence of attractor-repeller pairs for topological cocycles on compact spaces. The properties of attractor-repeller pairs and Morse decompositions are presented. [ABSTRACT FROM AUTHOR]

Precompact type properties - precompactness (=totally precompactness), s-precompactness, pre-Lindelöfness, (=ℵ0-boundedness), t -boundedness - belong to the basic important invariants studied in the uniform topology. The theory of these invariants is wide and continues to develop. However, in a sense, the class of uniformly Menger spaces escaped the attention of researchers. Lj.D.R. Kočinac was the first who introduced and studied the class of uniformly Menger spaces in [3, 4]. It immediately follows from the definition that the class of uniformly Menger spaces lies between the class of precompact uniform spaces and the class of pre-Lindelöf uniform spaces. Therefore, we expect it to have many good properties. In this paper some important properties of the uniformly Menger spaces are investigated. In particular, it is established that under uniformly perfect mappings, the uniformly Menger property is preserved both in the image and the preimage direction.

First, we introduce the concepts of equicontinuity, expansivity , ergodic shadowing property, and chain transitivity in uniform space. Second, we study the dynamical properties of equicontinuity, expansivity , ergodic shadowing property, and chain transitivity in the hyperspace of uniform space. Let (X,μ)\left(X,\mu ) be a uniform space, (C(X),Cμ)\left(C\left(X),{C}^{\mu }) be a hyperspace of (X,μ)\left(X,\mu ), and f:X→Xf:X\to X be uniformly continuous. By using the relationship between original space and hyperspace, we obtain the following results: (a) the map ff is equicontinous if and only if the induced map Cf{C}^{f} is equicontinous; (b) if the induced map Cf{C}^{f} is expansive, then the map ff is expansive; (c) if the induced map Cf{C}^{f} has ergodic shadowing property, then the map ff has ergodic shadowing property; (d) if the induced map Cf{C}^{f} is chain transitive, then the map ff is chain transitive. In addition, we also study the topological conjugate invariance of (G,h)\left(G,h)-shadowing property in metric GG- space and prove that the map SS has (G,h)\left(G,h)-shadowing property if and only if the map TT has (G,h)\left(G,h)-shadowing property. These results generalize the conclusions of equicontinuity, expansivity, ergodic shadowing property, and chain transitivity in hyperspace.

In this paper, we study the space S P G n X of G-permutation degree. Especially, new properties of this space are investigated. We study its metrazibility, semi-metrazibility, stratifiability, semi-stratifiability and uniformizability. Moreover, we study the uniform space (S P G n X , S P G n Υ) , investigating when this space is totally bounded (ω-bounded). Finally, a study of universal elements in classes of spaces that are presented in this article, and related open questions complete this paper. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of multi-sensitivity with respect to a vector within the context of non-autonomous dynamical systems on uniform spaces and provides insightful results regarding  N -sensitivity and strongly multi-sensitivity, along with their behaviors under various conditions. The main results established are as follows: (1) For a k-periodic nonautonomous dynamical system on a Hausdorff uniform space  (S , U) , the system  (S , f k ∘ ⋯ ∘ f 1)  exhibits  N -sensitivity (or strongly multi-sensitivity) if and only if the system  (S , f 1 , ∞)  displays  N -sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space  (S , U) . If the system  (S , f 1 , ∞)  is  N -sensitive, then the system  (S , f k , ∞)  is also  N -sensitive. When the family  f 1 , ∞  is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space  (S , U) ,  N -sensitivity (and strongly multi-sensitivity) persists under iteration. (4) We present a sufficient condition under which an nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates  N -sensitivity. [ABSTRACT FROM AUTHOR]

Shcherbakov's comparability method is very useful to study recurrent solutions of differential equations. In this paper, we extend the method from metric spaces to uniform spaces, which applies well to dynamical systems in infinitedimensional spaces. This generalized comparability method can be easily used to study recurrent solutions of (stochastic) partial differential equations under weaker conditions than in earlier results. We also show that the distribution of solutions of SDEs naturally generates a semiflow or skew-product semiflow on the space of probability measures, which is interesting in itself. As illustration, we give an application to semilinear stochastic partial differential equations. [ABSTRACT FROM AUTHOR]

We study topological ergodic shadowing, topological d ̲ shadowing and topological average shadowing property for a continuous map on a uniform space and show that they are equivalent for a uniformly continuous map with topological shadowing on a compact uniform space. Furthermore, we prove that topological average shadowing property with Lyapunov stability implies topological ergodicity. [ABSTRACT FROM AUTHOR]

In this paper, we focus on some properties, calculations and estimations of topological entropy for a nonautonomous dynamical system (X , f 0 , ∞) generated by a sequence of continuous self-maps f 0 , ∞ = { f n } n = 0 ∞ on a compact uniform space X. We prove that (X , f 0 , ∞) and its k -th product system have the same entropy. We confirm that the entropy of (X , f 0 , ∞) equals to that of its n -th compositions system, and also equals to the entropy of f 0 , ∞ restricted to its non-wandering set if f 0 , ∞ is equi-continuous. We prove that the entropy of (X , f 0 , ∞) is less than or equal to that of its limit system (X , f) when f 0 , ∞ converges uniformly to f. We show that two topologically equi-semiconjugate systems have the same entropy if the equi-semiconjugacy is finite-to-one. Finally, we obtain the estimations of upper and lower bounds of entropy for an invariant subsystem of a coupled-expanding system associated with a transition matrix. [ABSTRACT FROM AUTHOR]

The notion of dual uniformity is introduced on UC(Y,Z)UC(Y,Z), the uniform space of uniformly continuous mappings between YY and ZZ, where (Y,V)(Y,{\mathcal{V}}) and (Z,U)(Z,{\mathcal{U}}) are two uniform spaces. It is shown that a function space uniformity on UC(Y,Z)UC(Y,Z) is admissible (resp. splitting) if and only if its dual uniformity on UZ(Y)={f2−1(U)∣f∈UC(Y,Z),U∈U}{{\mathcal{U}}}_{Z}(Y)=\{{f}_{2}^{-1}(U)\hspace{0.33em}| \hspace{0.33em}f\in UC(Y,Z),U\in {\mathcal{U}}\} is admissible (resp. splitting). It is also shown that a uniformity on UZ(Y){{\mathcal{U}}}_{Z}(Y) is admissible (resp. splitting) if and only if its dual uniformity on UC(Y,Z)UC(Y,Z) is admissible (resp. splitting). Using duality theorems, it is also proved that the greatest splitting uniformity and the greatest splitting family open uniformity exist on UZ(Y){{\mathcal{U}}}_{Z}(Y) and UC(Y,Z)UC(Y,Z), respectively, and these two uniformities are mutually dual splitting uniformities.

This paper presents some variants of minimal point theorem together with corresponding variants of Ekeland variational principle. In the second part of this article, there is a discussion on Ekeland variational principle and minimal point theorem relative to it in uniform spaces. The aim of these series of important results is to highlight relations between them, some improvements and specific applications.

The classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the $ f $-statistical convergence for $ f $ a modulus function. Here, we go several steps further and extend the $ f $-statistical convergence to the scope of uniform spaces, obtaining particular cases of $ f $-statistical convergence on pseudometric spaces and topological modules.

In this paper we provide new characterizations of the Gehring-Hayman theorem from the point of view of Gromov boundary and uniformity. We also determine the critical exponents for the uniformized space to be a uniform space in the case of the hyperbolic spaces, the model spaces Mkn of the sectional curvature k < 0 with the dimension n ≥ 2 and hyperbolic fillings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several forms of transitivity and sensitivity in the dynamical system of semigroup actions. Firstly, we derive some equivalent conditions for a dynamical system to be sensitive. In particular, it is proved that an M-system with an equicontinuous point is minimal as well as equicontinuous, every syndetically transitive system with equicontinuous point is minimal as well as equicontinuous. Also, we derive equivalent conditions for a system to be almost equicontinuous in terms of sensitivity and equicontinuity. Besides, some results on point transitive and densely point transitive system on uniform spaces are obtained. [ABSTRACT FROM AUTHOR]

The classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the -statistical convergence for a modulus function. Here, we go several steps further and extend the -statistical convergence to the scope of uniform spaces, obtaining particular cases of -statistical convergence on pseudometric spaces and topological modules. [ABSTRACT FROM AUTHOR]

In this paper the properties of space of the G-permutation degree, like: weight, uniform connectedness and index boundedness are studied. It is proved that: (1) If (X, U) is a uniform space, then the mapping π s n, G : (X n , U n ) → (SP n GX, SP n GU) is uniformly continuous and uniformly open, moreover w (U) = w (SP n GU); (2) If the mapping f : (X, U) → (Y, V) is a uniformly continuous (open), then the mapping SP n Gf : (SP n GX, SP n GU) → (SP n GY, SP n GV) is also uniformly continuous (open); (3) If the uniform space (X, U) is uniformly connected, then the uniform space (SP n GX, SP n GU) is also uniformly connected.

We define notions such as pseudo-orbit, topological shadowing, and topological chain transitivity of iterated function systems on compact uniform spaces. We prove that these notions are invariant under topological conjugacy on a compact uniform space. For an IFS on a compact uniform space with topological shadowing property, we show that the topological chain transitivity implies topological transitivity. We also show that in a connected compact uniform space, notions such as topological chain mixing, totally topological chain transitive, topological chain transitive, and topological chain recurrent are equivalent. [ABSTRACT FROM AUTHOR]

We study here the topological notion of Li-Yorke chaos defined for uniformly continuous self-maps defined on uniform Hausdorff spaces, which are not necessarily compact metrizable. We prove that a weakly mixing uniformly continuous self-map defined on a second countable Baire uniform Hausdorff space without isolated points is Li-Yorke chaotic. Further, we define and study the notion of topological distributional chaos in a sequence for uniformly continuous self-maps defined on uniform Hausdorff spaces. We prove that Li-Yorke chaos is equivalent to topological distributional chaos in a sequence for uniformly continuous self-maps defined on second countable Baire uniform Hausdorff space without isolated points. As a consequence, we obtain that Devaney chaos implies topological distributional chaos in a sequence. [ABSTRACT FROM AUTHOR]

In the present paper, we establish that the functor of monetary risk measures with compact supports transforms open maps into open maps and preserves weight and completeness index of uniform spaces. Consequently, the space of monetary risk measures with compact support is locally compact Hausdorff space if and only if the original space is so. [ABSTRACT FROM AUTHOR]

In this paper, we relate the topological definitions of specification property and distributional chaos defined for uniformly continuous self-maps on noncompact, nonmetrizable spaces. We prove that a uniformly continuous surjective self-map acting on a uniformly locally compact Hausdorff uniform space with topologically weak specification property and a pair of distal points is topologically distributionally chaotic of type 1. This extends the result due to Oprocha and Štefánková [2008]. As a consequence, we get that uniformly continuous surjective self-map on a uniformly locally compact totally bounded Hausdorff uniform space with topological shadowing, topological mixing, and a distal pair is topologically distributionally chaotic of type 1. [ABSTRACT FROM AUTHOR]

New families of uniformities are introduced on UC(X,Y)UC(X,Y), the class of uniformly continuous mappings between X and Y, where (X,U)(X,{\mathcal{U}}) and (Y,V)(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.

Given a Hausdorff uniform space with the countable gage of pseudometrics of the uniformity of , we introduce a concept of the approximate variation of a function mapping a subset of the reals into : this is the infimum of the family of Jordan-type variations of all functions which differ from in each uniform pseudometric, generated by a pseudometric from the gage, not greater than . We prove the following compactness theorem in the topology of pointwise convergence: if a pointwise relatively sequentially compact sequence of functions is such that the limit superior of its approximate variations is finite for all pseudometrics in the gage and all , then it contains a subsequence which converges pointwise on the domain to a bounded regulated function (in a generalized sense). We illustrate this result by appropriate sharp examples and present a new characterization of uniform space valued regulated functions in terms of the approximate variation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces. We introduce the concept of topological stability for homeomorphisms on compact uniform spaces and prove that if a homeomorphism on a compact uniform space is expansive and has pseudo-orbit tracing property, then it is topologically stable. Moreover, we discuss the topological stability for homeomorphisms on uniform spaces from the view of localization. We introduce definitions of topologically stable point and shadowable point for homeomorphisms on uniform spaces and show that every shadowable point of an expansive homeomorphism on a compact uniform space is topologically stable. [ABSTRACT FROM AUTHOR]

In this paper, the connection between a finally compact, pceudocompact, extremely disconnected, -space and its hyperspace is studied. The action of functors on finally compact, pceudocompact, extremely disconnected and -spaces is investigated. Some topological properties of uniformly space and its hyperspace is studied. It is proved: if the uniform space is uniformly paracompact, then is uniformly paracompact. It is also shown: if the uniform space is uniformly -paracompact, then a uniform space is uniformly -paracompact. [ABSTRACT FROM AUTHOR]

We consider the relationship between normality and semi-proximality. We give a consistent example of a first countable locally compact Dowker space that is not semi-proximal, and two ZFC examples of semi-proximal non-normal spaces. This answers a question of Nyikos. One of the examples is a subspace of (ω + 1) × ω 1. In contrast, we show that every normal subspace of a finite power of ω 1 is semi-proximal. [ABSTRACT FROM AUTHOR]

A uniform space X is said to be proximally fine if every proximally continuous function defined on X into an arbitrary uniform pace Y is uniformly continuous. We supply a proof that every topological group which is functionally generated by its precompact subsets is proximally fine with respect to its right uniformity. On the other hand, we show that there are various permutation groups G on the integers N that are not proximally fine with respect to the topology generated by the sets {g ∈ G : g(A) ⊂ B}, A, B ⊂ N.

n this paper, We present some important dynamical concepts of uniform space such as the uniform minimal systems and uniform shadowing. We explain some definitions and theorems such as uniform expansive, weak uniform expansive, uniform generator. we study the relations among types of compact uniform space and uniform shadowing as well as uniform conjugacy. Moreover we show that if T: X⟶ Χ and S: Y → Y are two uniform homeomorphisms on compact uniform spaces X and Y, if φ: X ➞ Y is a uniform conjugacy from T to S, then CRU(S) = φ(CRU(T)). [ABSTRACT FROM AUTHOR]

We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable. [ABSTRACT FROM AUTHOR]

We introduce topological definition of average shadowing property. We prove that topological average shadowing property implies topological chain transitivity. In particular it is proved that for a dynamical system with dense minimal points, the topological average shadowing property implies topological strong ergodicity. [ABSTRACT FROM AUTHOR]

In this paper the properties of space of the G-permutation degree, like: weight, uniform connectedness and index boundedness are studied. It is proved that: (1) If (X, U) is a uniform space, then the mapping s n,G: (Xn, Un) ! (SPn GX, SPn GU) is uniformly continuous and uniformly open, moreover w (U) = w (SPn GU); (2) If the mapping f: (X, U) ! (Y, V) is a uniformly continuous (open), then the mapping SPn Gf: (SPn GX, SPn GU) ! (SPn GY, SPn GV) is also uniformly continuous (open); (3) If the uniform space (X, U) is uniformly connected, then the uniform space (SPn GX, SPn GU) is also uniformly connected. [ABSTRACT FROM AUTHOR]

In the study of aperiodic order via dynamical methods, topological entropy is an important concept. In this paper, parts of the theory, like Bowen's formula for fibre wise entropy or the independence of the definition from the choice of a Van Hove sequence, are extended to actions of several non-discrete groups. To establish these results, we will show that the Ornstein–Weiss lemma is valid for all considered groups which appear in the study of cut and project schemes. [ABSTRACT FROM AUTHOR]

The uniformization and hyperbolization transformations formulated by Bonk et al. in "Uniformizing Gromov Hyperbolic Spaces", Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p-Poincaré inequality, then the transformed measure is globally doubling and supports a global p-Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p-Poincaré inequality, carries nonconstant globally defined p-harmonic functions with finite p-energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain. [ABSTRACT FROM AUTHOR]

Function space topologies are developed for EC(Y,Z), the class of equi-continuous mappings from a topological space Y to a uniform space Z. Prop-erties such as splittingness, admissibility etc. are defined for such spaces. The net theoretic investigations are carried out to provide characteriza-tions of splittingness and admissibility of function spaces on EC(Y,Z). The open-entourage topology and pointtransitive-entourage topology are shown to be admissible and splitting respectively. Dual topologies are de-fined. A topology on EC(Y,Z) is found to be admissible (resp. splitting) if and only if its dual is so. [ABSTRACT FROM AUTHOR]

In this paper we introduce ζ-contraction via simulation function in uniform spaces and prove new fixed point theorem for ζ-contraction in uniform spaces. [ABSTRACT FROM AUTHOR]

In this paper we establish an existence result for fixed points of mapping in a uniform space, which extends some previous theorems of V. G. Angelov [1].

We introduce the concepts of sensitivity, multisensitivity, cofinite sensitivity, and syndetic sensitivity for nonautonomous dynamical systems on uniform spaces and obtain some sufficient conditions under which topological transitivity and dense periodic points imply sensitivity for nonautonomous systems on Hausdorff uniform spaces. We also study sensitivity and other stronger versions of sensitivity for the systems induced on hyperspaces and for the product of nonautonomous dynamical systems on uniform spaces. [ABSTRACT FROM AUTHOR]

In the present paper we established that the functor of idempotent probability measures with compact supports transforms perfect maps into perfect maps and preserves weight and completeness index of uniform spaces. [ABSTRACT FROM AUTHOR]

The paper is an overview of selected results on weaker forms of classical selection principles of Menger, Hurewicz, Rothberger and Gerlits-Nagy obtained in the last few years. We review some very recent results on star selection principles, an important subfield of selection principles theory (including boundedness properties in function spaces considered as uniform spaces and topological groups), ideal versions of Hurewicz-type properties, selection principles called weak, almost, quasi, and combinatorics of covers by generalized open sets. A number of examples which clarify relationships among properties we consider is given, and some open questions are posed. [ABSTRACT FROM AUTHOR]

We consider the notions of equicontinuity point, sensitivity point and so on from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. We show that for the notions of equicontinuity point and sensitivity point, Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definitions stated in terms of a metric in compact metric spaces. We prove that a uniformly chain transitive map with uniform shadowing property on a compact Hausdorff uniform space is either uniformly equicontinuous or it has no uniform equicontinuity points. [ABSTRACT FROM AUTHOR]

In this paper we extend some results on the notions such as Expansiveness, Pseudo Orbit Tracing Property (P.O.T.P.), Chain Transitivity, Periodic Shadowing property to uniform dynamical system (X; f), where X is a compact uniform space. We prove that if an expansive dynamical system (X; f) on compact uniform space has P.O.T.P., then it has periodic shadowing. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce and study here the notion of distributional chaos on uniform spaces. We prove that if a uniformly continuous self-map of a uniform locally compact Hausdorff space has topological weak specification property then it admits a topologically distributionally scrambled set of type 3. This extends result due to Sklar and Smítal (J Math Anal Appl 241:181–188, 2000). We also justify through examples necessity of the conditions in the hypothesis of the main result. [ABSTRACT FROM AUTHOR]