Recent zbMATH articles in MSC 54Chttps://zbmath.org/atom/cc/54C2024-09-27T17:47:02.548271ZWerkzeugEquicontinuity and sensitivity of group actionshttps://zbmath.org/1541.370072024-09-27T17:47:02.548271Z"Xie, Shaoting"https://zbmath.org/authors/?q=ai:xie.shaoting"Yin, Jiandong"https://zbmath.org/authors/?q=ai:yin.jiandongSummary: Let \((X, G)\) be a dynamical system \((G\)-system for short), that is, \(X\) is a topological space and \(G\) is an infinite topological group continuously acting on \(X\). In the paper, the authors introduce the concepts of Hausdorff sensitivity, Hausdorff equicontinuity and topological equicontinuity for \(G\)-systems and prove that a minimal \(G\)-system \((X, G)\) is either topologically equicontinuous or Hausdorff sensitive under the assumption that \(X\) is a \(T_3\)-space and they provide a classification of transitive dynamical systems in terms of equicontinuity pairs. In particular, under the condition that \(X\) is a Hausdorff uniform space, they give a dichotomy theorem between Hausdorff sensitivity and Hausdorff equicontinuity for \(G\)-systems admitting one transitive point.Topological stability and entropy for certain set-valued mapshttps://zbmath.org/1541.370122024-09-27T17:47:02.548271Z"Zhang, Yu"https://zbmath.org/authors/?q=ai:zhang.yu.130|zhang.yu.189|zhang.yu.143|zhang.yu.158|zhang.yu.29|zhang.yu.32|zhang.yu.99|zhang.yu.95|zhang.yu.66|zhang.yu.13|zhang.yu.27|zhang.yu.73|zhang.yu.167|zhang.yu.12|zhang.yu.38|zhang.yu.58|zhang.yu.172|zhang.yu.77|zhang.yu.74|zhang.yu.148|zhang.yu.8|zhang.yu.142|zhang.yu.15|zhang.yu.144|zhang.yu.69|zhang.yu.82|zhang.yu.70|zhang.yu.153|zhang.yu.134|zhang.yu.181|zhang.yu.4|zhang.yu.173|zhang.yu.121|zhang.yu.157|zhang.yu.101|zhang.yu.19|zhang.yu.178|zhang.yu.154|zhang.yu.177|zhang.yu.170|zhang.yu.35|zhang.yu.88|zhang.yu.180|zhang.yu.20"Zhu, Yu Jun"https://zbmath.org/authors/?q=ai:zhu.yujunSummary: In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if \(f\) is a hyperbolic endomorphism then for each \(\varepsilon > 0\) there exists a \(C^1\)-neighborhood \(\mathcal{U}\) of \(f\) such that the induced set-valued map \(F_{f, \mathcal{U}}\) has the \(\varepsilon\)-shadowing property, and moreover, if \(f\) is an expanding endomorphism then there exists a \(C^1\)-neighborhood \(\mathcal{U}\) of \(f\) such that the induced set-valued map \(F_{f, \mathcal{U}}\) has the Lipschitz shadowing property; (2) when a set-valued map \(F\) is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then \(F\) is expansive and hence is topologically stable; (3) if \(f\) is an expanding endomorphism then for each \(\varepsilon > 0\) there exists a \(C^1\)-neighborhood \(\mathcal{U}\) of \(f\) such that \(h (F_{f, \mathcal{U}}, \varepsilon) = h(f)\); (4) when \(F\) is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of \(F\) is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.Metric mean dimension via preimage structureshttps://zbmath.org/1541.370132024-09-27T17:47:02.548271Z"Liu, Chunlin"https://zbmath.org/authors/?q=ai:liu.chunlin"Rodrigues, Fagner B."https://zbmath.org/authors/?q=ai:rodrigues.fagner-bLet \((X, d)\) be a compact metric space and \(f : X\to X\) be a continuous map. \textit{M. Hurley} [Ergodic Theory Dyn. Syst. 15, No. 3, 557--568 (1995; Zbl 0833.54021)] introduced the topological preimage entropy of \(f\) as follows
\[
h_m(f)=\lim_{\varepsilon \rightarrow 0} h_m(f,\varepsilon ) =\underset{n\rightarrow \infty }{\lim \sup }\frac{1}{n}\log \underset{x\in X}{\sup }s(n,\varepsilon ,f^{-n}(x)), \] where \(s(n,\varepsilon,f^{-n}(x))\) is the maximal cardinality of all \((n,\varepsilon)\)-separated subsets of \(f^{-n}(x)\) with respect to \(f\). Another potential definition of preimage entropy, proposed by \textit{W.-C. Cheng} and \textit{S. E. Newhouse} [Ergodic Theory Dyn. Syst. 25, No. 4, 1091--1113 (2005; Zbl 1098.37012)], is the following
\[
h_{pre}(f)=\lim_{\varepsilon \rightarrow 0} h_{pre}(f,\varepsilon)=\underset{n\rightarrow \infty }{\lim \sup }\frac{1}{n}\log\underset{x\in X,k\geq n}{\sup }s(n,\varepsilon ,f^{-k}(x)).
\]
In this paper, inspired by earlier definitions of preimage entropy, the authors propose two concepts of upper and lower metric mean dimension based on preimage structures. The authors investigate the measure-theoretic preimage entropy and the preimage metric mean dimension. Inspired by previous studies, they estimate new upper and lower bounds depending on whether the function \(f\) is continuous and surjective. Several intermediate results are given to prove the main theorems.
Reviewer: Hasan Akin (Şanlıurfa)Shadowing, transitivity and a variation of omega-chaoshttps://zbmath.org/1541.370182024-09-27T17:47:02.548271Z"Kawaguchi, Noriaki"https://zbmath.org/authors/?q=ai:kawaguchi.noriakiSummary: We study a special type of shadowing (DSP) of chain transitive continuous self-maps of compact Hausdorff spaces. We prove some basic properties of DSP. As application of DSP, we obtain sufficient conditions for a statistical variant of \(\omega\)-chaos and prove the topological genericity of it. We also consider topological distribution of irregular points under the assumption of DSP.Dynamics of iteration operators on self-maps of locally compact Hausdorff spaceshttps://zbmath.org/1541.390192024-09-27T17:47:02.548271Z"Gopalakrishna, Chaitanya"https://zbmath.org/authors/?q=ai:gopalakrishna.chaitanya"Veerapazham, Murugan"https://zbmath.org/authors/?q=ai:veerapazham.murugan"Zhang, Weinian"https://zbmath.org/authors/?q=ai:zhang.weinianThe authors show the continuity of iteration operators \(\mathcal{J}_{n}\) on the space of all continuous self-maps of a locally compact Hausdorff space \(X\) and discuss their dynamical behavior. In the beginning, they give some preliminaries for locally compact Hausdorff spaces, which are used in the proofs of their results. The authors firstly discuss the dynamical behavior of the semi-dynamical system \((\mathcal{C}(X), \mathcal{J}_{n})\), where \(X\) is a locally compact Hausdorff space, by giving the continuity of \(\mathcal{J}_{n}\) on \(\mathcal{C}(X)\).
The authors characterize the fixed points and the periodic points of the considered systems, also
by considering the Babbage equation for \(X= \mathbb{R}\) and the unit circle \(X=\mathbb{S}^{1}\). They note that in \(\mathcal{C}(\mathbb{S}^{1})\) each \(\mathcal{J}_{n}\) may have periodic points of period \(k \geq 2\), whereas in \(\mathcal{C}(\mathbb{R})\), each \(\mathcal{J}_{n}\) only has fixed points. It is proved that if \(f\) is a strictly monotonic fixed point of \(\mathcal{J}_{3}\) in \(\mathcal{C}(\mathbb{R})\), then either \(f = \text{id}\) or \(f\) is a strictly decreasing involutory function on \(\mathbb{R}\). Moreover, for each \(n \in \mathbb{N}\), \(\mathcal{J}_{n}\) does not have periodic points of period \(k \geq 2\) in \(\mathcal{C}(\mathbb{R})\).
Further, the authors describe the stability for \(\mathcal{J}_{n}\). They explain that all orbits of \(\mathcal{J}_{n}\) are bounded; however, it is proved that for \(X= \mathbb{R}\) and \(\mathbb{S}^{1}\), every fixed point of \(\mathcal{ J}_{n}\) which is non-constant and equals the identity on its range is not Lyapunov stable. The boundedness and the instability exhibit the complexity of the system, but the authors observe that the complex behavior is not Devaney chaotic. The authors give here some examples to justify their results.
Then they provide a sufficient condition to classify the systems generated by iteration of operators up to topological conjugacy. Let \(X\) and \(Y\) be locally compact Hausdorff spaces. It is shown that if \(X\) is homeomorphic to \(Y\), then \((\mathcal{C}(X), \mathcal{J}_{n})\) is conjugate to\((\mathcal{C}(Y),\mathcal{J}_{n})\) for each \(n \in \mathbb{N}\). Further, they show that \((\mathcal{C}(X), \mathcal{J}_{2})\) is not conjugate to \((\mathcal{C}(Y), \mathcal{J}_{m})\) for every locally compact Hausdorff space \(Y\) and odd positive integer \(m\) whenever \(\mathcal{C}(X)\) contains an involutory map different from id.
At the end of the paper, the authors provide some remarks and list some problems for future work.
Reviewer: Mohammad Sajid (Buraydah)Pure metric geometryhttps://zbmath.org/1541.540012024-09-27T17:47:02.548271Z"Petrunin, Anton"https://zbmath.org/authors/?q=ai:petrunin.antonThis is a succinct introduction to the tools one needs to understand metric geometry. It starts with the most basic definitions, such as that of a metric space, of completeness (with a proof of the Baire category theorem), of a compact space, of a geodesic, of metric trees, or length spaces (with a proof of Menger's lemma and of the Hopf-Rinow theorem), to move on to more advanced topics, such as the extension property, universality, the Urysohn space, injective spaces, the Hausdorff metric, the Gromov-Hausdorff metric, and ultrafilters. There are exercises of various levels of difficulty with ``semisolutions'' at the back of the book. For complicated solutions, the source of the solution in the literature is also mentioned.
Reviewer: Victor V. Pambuccian (Glendale)Binary soft continuous functions and their characterizationshttps://zbmath.org/1541.540022024-09-27T17:47:02.548271Z"Patil, P. G."https://zbmath.org/authors/?q=ai:patil.prakashgouda-g|patil.parag-g|patil.prakash-g"Adaki, Asha G."https://zbmath.org/authors/?q=ai:adaki.asha-gSummary: The current study is concerned with the concept of binary soft semi-open sets and binary soft somewhat-open sets. Later, binary soft continuous functions, binary soft semi-continuous functions, and binary soft somewhat-continuous functions were defined and investigated interrelation between them.On Lipschitz partitions of unity and the Assouad-Nagata dimensionhttps://zbmath.org/1541.540092024-09-27T17:47:02.548271Z"Licht, Martin W."https://zbmath.org/authors/?q=ai:licht.martin-wernerLet \(X\) be a metric space, and its metric is written \(\Delta\). Given an open cover \(\mathcal{U}=(U_{\alpha})_{\alpha\in \kappa}\) of a metric space \(X\), let \(\delta_{\alpha}: X\to \mathbb{R}\) be the map defined by \(\delta_{\alpha}(x)=\Delta (x, X\setminus U_{\alpha})\), \(\alpha \in \kappa\), \(x\in X\). The functions
\[
\lambda_{\alpha}(x)= \frac{\delta_{\alpha}(x)}{\sum_{\beta\in \kappa} \delta_{\beta}(x)},~\alpha \in \kappa
\]
constitute a continuous partition of unity \(\Lambda =(\lambda_{\alpha})_{\alpha\in \kappa}\) subordinate to \(\mathcal{U}\). This \(\Lambda\) is called the \textit{standard partition of unity subordinate to} \(\mathcal{U}\). We say that \(X\) satisfies the \textit{approximate midpoint property} if for all \(x,y\in X\) and \(\epsilon>0\) there exists \(z\in X\) such that
\[
\max (\Delta (x,z), \Delta(z,y))\leq \Delta(x,y)/2 +\epsilon.
\]
The author in the paper under review shows that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant \(\max (1, M-1)/\mathcal{L}\), where \(\mathcal{L}\) is the Lebesgue number and \(M\) is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, then the upper bound improves to \((M-1)/(2\mathcal{L})\). Additionally, the Lipschitz constants of partitions of unity modeled on more general \(\ell^{p}\) normalization are also estimated (Theorem 3.4). Finally, the author characterizes metric spaces with Assouad-Nagata dimension \(n\) as precisely those metric spaces for which every Lebesgue cover admits an open refinement with multiplicity \(n+1\) while reducing the Lebesgue number by at most a constant factor (Proposition 4.5).
Reviewer: Yutaka Iwamoto (Niihama)On \(e\)-spaces and rings of real valued \(e\)-continuous functionshttps://zbmath.org/1541.540102024-09-27T17:47:02.548271Z"Afrooz, S."https://zbmath.org/authors/?q=ai:afrooz.soosan|afrooz.susan"Azarpanah, F."https://zbmath.org/authors/?q=ai:azarpanah.f"Hajee, N. Hasan"https://zbmath.org/authors/?q=ai:hajee.n-hasanA subset \(Y\) of a topological space \((X,\tau)\) is said to be \textit{\(e\)-open} if the closure of \(Y\) is an open set in \(X\). In this interesting paper, the authors show that the collection of all \(e\)-open sets forms a base for some topology on \(X\). If the new topology coincides with the original topology, then \(X\) is called an \(e\)-space, a new addition to topological invariants. In the spirit of topology, they build new topological spaces, coming from various separation axioms subject to \(e\)-open sets. In that process, they observe that the class of \(T_3\)-\(e\)-spaces and the class of zero-dimensional spaces coincide, and if \(X\) is an \(e\)-space then \(X\) is Hausdorff if and only if it is ultra Hausdorff.
In the following section, the authors define \(e\)-continuous functions which generalize the concept of clopen continuous functions, introduced in [\textit{S. Afrooz} et al., Appl. Gen. Topol. 19, No. 2, 203--216 (2018; Zbl 1402.54025)]. It is shown that an \(e\)-continuous image of an \(e\)-compact space is again \(e\)-compact, this resonates with the classical case, that is, a continuous image of a compact space is compact.
Denote by \(C_e(X)\) the ring of real valued \(e\)-continuous functions on \(X\). The ring \(C_e(X)\) turns out to be a \(C\)-ring, that is, isomorphic to \(C(Y)\) for a zero-dimensional space \(Y\) whose elements are the quasicomponents of \(X\).
The final section is focused on maximal ideals of \(C_e(X)\). The authors obtain interesting results and achieve a complete characterization of maximal ideals and real maximal ideals of \(C_e(X)\). An upshot of this section is a new characterization of zero-dimensional spaces which they achieve via the weak topology induced by \(C_e(X)\) in the presence of Hausdorffness on \(X\).
Reviewer: Sagarmoy Bag (Tangrakhali)A dynamical approach to shapehttps://zbmath.org/1541.540112024-09-27T17:47:02.548271Z"Shoptrajanov, Martin"https://zbmath.org/authors/?q=ai:shoptrajanov.martinThe main goal of this article is to get closer to answering the open question: ``Let \(X\) be a compact metric space and \(\left\lbrace\Psi_t:X\to X:t\in\mathbb{R}^+\right\rbrace\) a semiflow with a global attractor \(M\). Is every shape equivalence \(f:M\to X\) a strong shape equivalence?'' Although this question is purely theoretical and belongs to shape theory, in this article the author tries to ``prepare the ground'' to facilitate the solution of the aforementioned problem using dynamical tools.\newline
For this purpose, Theorem 4.3 is stated and proved in detail, which states that the inclusion \(i:M\to X\) induces a strong shape equivalence from a global attractor \(M\) to its phase space \(X\). This theorem gives a new insight of the already mentioned question. Namely, one can state the following problem which is purely dynamical by nature: ``Given a set \(X\) in a mapping cylinder \(M\left(f\right)\) such that the inclusion \(i:X\to M\left(f\right)\) induces a shape equivalence, when is \(X\) the global attractor for some semi-dynamical system in \(M\left(f\right)\)?'' Attacking this new dynamical problem one can actually obtain, at least partially, an answer to the purely shape theoretical problem.\newline
Finally, Theorem 4.9, which is stated and proved, builds up nicely to the Theorems 4.3 and 4.4 and gives new facts about the spaces of components of the global attractor \(M\).
Reviewer: Ivan Jelić (Split)The Hurewicz property and the Vietoris hyperspacehttps://zbmath.org/1541.540122024-09-27T17:47:02.548271Z"Caruvana, Christopher"https://zbmath.org/authors/?q=ai:caruvana.christopherDenote by \(\mathbb K(X)\) (resp. \(\mathcal P_{\mathrm{fin}}(X)\)) the space of non-empty compact (resp. non-empty finite) subsets of a given topological space \(X\) with the Vietoris topology. It is shown that \(\mathbb K(X)\) is Hurewicz iff \(X\) (equivalently \(\mathbb K(X)\)) satisfies a certain finite selection principle iff a specified player does not have a winning strategy in a given game. Further, \(\mathcal{P}_{\mathrm{fin}}(X)\) is Hurewicz iff every finite power of \(X\) is Hurewicz, with equivalent conditions involving a selection principle and lack of a winning strategy in a game.
Reviewer: David B. Gauld (Auckland)How much can we extend the Assouad embedding theorem?https://zbmath.org/1541.540142024-09-27T17:47:02.548271Z"Turoboś, Filip"https://zbmath.org/authors/?q=ai:turobos.filip"Dovgoshey, Oleksiy"https://zbmath.org/authors/?q=ai:dovgoshey.oleksiy-aA semimetric space \((X,d)\) is a b-metric space if there is \(K\ge1\) such that \(d(x,z)\le K(d(x,y)+d(y,z))\) for each \(x,y,z\in X\), and is doubling if there is \(m\in\mathbb N\) such that for any \(x\in X\) and \(r>0\) the open ball \(B_d(x;r)\) can be covered by at most \(m\) many open balls of radius \(\frac{r}{2}\). Suppose that \((X,d)\) is a b-metric space with the doubling property. Then there is \(n\in\mathbb N\) such that for all \(\alpha\in(0,1)\) there are a constant \(C\) and injective \(F:X\to\mathbb R^n\) such that \(C^{-1}d^\alpha(x,y)\le\|F(x)-F(y)\|\le Cd^\alpha(x,y)\) for all \(x,y\in X\). A partial converse, viz if \((X,d)\) is a semimetric space such that for some \(\alpha\in(0,1]\) there is a bi-Lipschitz embedding of \((X,d^\alpha)\) into a metric space \((Y,\rho)\) then \((X,d)\) is a b-metric space, is also proved.
For the entire collection see [Zbl 1518.26001].
Reviewer: David B. Gauld (Auckland)\(K\)-theory of co-existentially closed continuahttps://zbmath.org/1541.540182024-09-27T17:47:02.548271Z"Eagle, Christopher J."https://zbmath.org/authors/?q=ai:eagle.christopher-james|eagle.christopher-j"Lau, Joshua"https://zbmath.org/authors/?q=ai:lau.joshuaThe authors describe the \(K\)-theory of co-existentially closed continua. A continuum, \(X\), is said to be co-existentially closed if it is dual to an existentially closed structure: one can use Wallman bases for the closed sets (first-order logic) or the \(C^*\)-algebra of complex-valued continuous functions (continuous logic). The paper uses the latter approach, and describes \(K_0(C(X))\) and \(K_1(C(X))\).\par If \(X\) is a one-dimensional continuum then \(K_0(C(X))=\mathbb{Z}\); as co-existentially closed continua are one-dimensional this determines their~\(K_0\).
Again, if \(X\) is a one-dimensional continuum then \(K_1(C(X))\) is a known object: it is the first integral Čech cohomology group \(\check{H}^1(X)\). The authors show that for co-existentially closed continua the latter group is a divisible and torsion-free abelian group. They then go on to show that for every infinite cardinal~\(\kappa\) there is a co-existentially closed continuum of weight~\(2^\kappa\) for which the group \(\check{H}^1(X)\) has rank at least \(2^\kappa\).\par As an application the authors exhibit a large family of metrizable continua that are not co-existentially closed: they are circle-like, not arc-like, and hereditarily indecomposable -- called pseudosolenoids.
Reviewer: K. P. Hart (Delft)Cardinal-indexed classifying spaces for families of subgroups of any topological grouphttps://zbmath.org/1541.550152024-09-27T17:47:02.548271Z"Khan, Qayum"https://zbmath.org/authors/?q=ai:khan.qayumLet \(G\) be a topological group. The author generalizes existence theorems by \textit{J. W. Milnor} [Ann. Math. (2) 63, 430--436 (1956; Zbl 0071.17401)], \textit{I. M. Gel'fand} and \textit{D. B. Fuks} [Sov. Math., Dokl. 9, 851--854 (1968; Zbl 0181.26602); translation from Dokl. Akad. Nauk SSSR 181, 515--518 (1968)], and \textit{G. B. Segal} [Funct. Anal. Appl. 9, 131--133 (1975; Zbl 0317.55017); translation from Funkts. Anal. Prilozh. 9, No. 2, 48--50 (1975)] of classifying spaces for principal \(G\)-bundles to \(G\)-spaces with torsion. For \(G\) a Lie group, via a metric model, the corresponding uniqueness theorem by \textit{R. S. Palais} [The classification of \(G\)-spaces. Providence, RI: American Mathematical Society (AMS) (1960; Zbl 0119.38403)] and \textit{G. E. Bredon} [Introduction to compact transformation groups. New York, NY: Academic Press (1972; Zbl 0246.57017)] are generalized for compact \(G\). The former existence result is enabled by Segal's clever but esoteric use of non-Hausdorff spaces. The latter uniqueness result is enabled by the author's own development of equivariant \(ANR\) theory for non-compact Lie \(G\).
Applications include the existence part of classification for unstructured fiber bundles with locally compact Hausdorff fiber and with locally connected base or fiber, as well as for equivariant principal bundles which in certain cases via other models is due to \textit{R. K. Lashof} and \textit{J. P. May} [Bull. Soc. Math. Belg., Sér. A 38, 265--271 (1986; Zbl 07894228)] and to \textit{W. Lück} and \textit{B. Uribe} [Algebr. Geom. Topol. 14, No. 4, 1925--1995 (2014; Zbl 1307.55008)]. From a categorical perspective, the presented general model \(E^\kappa_\mathcal{F}G\) is a final object inspired by the formulation of the Baum-Connes conjecture [\textit{P. Baum} et al., Contemp. Math. 167, 241--291 (1994; Zbl 0830.46061)].
Reviewer: Marek Golasiński (Olsztyn)