Metrization theorem
View article: What is topological about topological dynamics?
What is topological about topological dynamics? Open
We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions c…
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Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem Open
For $G$ a closed subgroup of $S_{\infty }$, we provide a precise combinatorial characterization of when the universal minimal flow $M(G)$ is metrizable. In particular, each such instance fits into the framework of metrizable flows develope…
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Metrizable universal minimal flows of Polish groups have a comeagre\n orbit Open
We prove that, whenever $G$ is a Polish group with metrizable universal\nminimal flow $M(G)$, there exists a comeagre orbit in $M(G)$. It then follows\nthat there exists an extremely amenable, closed, coprecompact $G^*$ of $G$ such\nthat $…
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Contractive function systems, their attractors and metrization Open
In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural conne…
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Topology and dynamics of the contracting boundary of cocompact CAT(0) spaces Open
Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the c…
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Completeness of the isomorphism problem for separable C*-algebras Open
This paper studies the descriptive set-theoretical complexity of the isomorphism problem for separable C*-algebras. We prove that the isomorphism problem for separable (simple, AI) C*-algebras is complete in the class of orbit equivalence …
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Localization of directed polymers in continuous space Open
The first main goal of this article is to give a new metrization of the Mukherjee–Varadhan topology, recently introduced as a translation-invariant compactification of the space of probability measures on Euclidean spaces. This new metriza…
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Extended suprametric spaces and Stone-type theorem Open
Extended suprametric spaces are defined, and the contraction principle is established using elementary properties of the greatest lower bound instead of the usual iteration procedure. Thereafter, some topological results and the Stone-type…
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Decomposing Borel functions using the Shore–Slaman join theorem Open
Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_\sigma $ set under tha…
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UNIVERSAL MINIMAL FLOWS OF GENERALIZED WAŻEWSKI DENDRITES Open
We study universal minimal flows of the homeomorphism groups of generalized Ważewski dendrites W P , $P \subseteq \left\{ {3,4, \ldots ,\omega } \right\}$ . If P is finite, we prove that the universal minimal flow of the homeomorphism grou…
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On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero Open
We prove that a Hausdorff locally compact semitopological bicyclic semigroup with adjoined zero $\mathscr{C}^0$ is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological bicyclic sem…
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Shadowing, ergodic shadowing and uniform spaces Open
We introduce and study the topological concepts of ergodic shadowing, chain transitivity and topological ergodicity for dynamical systems on non-compact non-metrizable spaces. These notions generalize the relevant concepts for metric space…
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Only 3-generalized metric spaceshave a compatible symmetric topology Open
We prove that every 3-generalized metric space is metrizable. We also show that for any ʋ with ʋ ≥ 4, not every ʋ-generalized metric space has a compatible symmetric topology.
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On the topology of fuzzy metric type spaces Open
In this paper, first we introduce the concept of fuzzy metric type space and consider the topology induced by a fuzzy metric type. Next, we consider the complete fuzzy metric type spaces and prove that any G? set in a complete metric type …
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Bernoulli disjointness Open
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On continuous self-maps and homeomorphisms of the Golomb space Open
The Golomb space ${\mathbb N}_\tau$ is the set ${\mathbb N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+ bn: n\ge 0\}$ with coprime $a,b$. We prove that the Golomb…
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On countably uniformly paracompact spaces Open
In this work we introduce and study countably uniformly paracompact spaces. In particular, the problem is solved: what are uniform spaces which for any finitely additive countable open covering ω admit a uniformly continuous ω-mapping to s…
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A Study of Boundedness in Fuzzy Normed Linear Spaces Open
In the present paper some different types of boundedness in fuzzy normed linear spaces of type ( X , N , ∗ ) , where ∗ is an arbitrary t-norm, are considered. These boundedness concepts are very general and some of them have no corresponde…
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Metric Compactifications and Coarse Structures Open
Let TB be the category of totally bounded, locally compact metric spaces with the C 0 coarse structures. We show that if X and Y are in TB , then X and Y are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact…
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Locally Quasi-Convex Compatible Topologies on a Topological Group Open
For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obvious…
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On the Structure of Unique Shortest Paths in Graphs Open
This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a grap…
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Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise Open
We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset of the parameterized space such that the bin…
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Pure point spectrum for dynamical systems and mean, Besicovitch and Weyl almost periodicity Open
We consider metrizable ergodic topological dynamical systems over locally compact, $\sigma $ -compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More spec…
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Extended Fuzzy Metrics and Fixed Point Theorems Open
In this paper, we study those fuzzy metrics M on X, in the George and Veeramani’s sense, such that ⋀ t > 0 M ( x , y , t ) > 0 . The continuous extension M 0 of M to X 2 × 0 , + ∞ is called extended fuzzy metric. We prove that M 0 generate…
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New Fixed Point Tools in Non-metrizable Spaces Open
The aim of this paper is to provide some sufficient conditions under which a self-mapping T defined on a non-empty set X endowed with some convergence property is a Picard operator. A relevant example showing that such a mapping T on a non…
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Dynamics of shift operators on non-metrizable sequence spaces Open
[EN] We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical …
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Minimal Spaces with Cyclic Group of Homeomorphisms Open
There are two main subjects in this paper. (1) For a topological dynamical system $$(X,T)$$ we study the topological entropy of its "functional envelopes" (the action of $$T$$ by left composition on the space of all continuous self-maps or…
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Weighted partition of a compact metrizable space, its hyperbolicity and Ahlfors regular conformal dimension Open
Successive divisions of compact metric spaces appear in many different areas of mathematics such as the construction of self-similar sets, Markov partitions associated with hyperbolic dynamical systems, dyadic cubes associated with a doubl…
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Several results on compact metrizable spaces in $$\mathbf {ZF}$$ Open
In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ , some are shown to be independent of $$\…
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Spaces of Continuous Functions Open
This chapter is dedicated to the properties of spaces of continuous functions taking values in a semi-normed space. It presents the definitions of the space of continuous functions, the space of uniformly continuous functions, and variants…