Pub Date : 2024-11-05DOI: 10.1016/j.topol.2024.109130
Nikola Bogdanovic
In recent years, Barbieri, Dikranjan, Giordano Bruno and Weber have made progress on the problem of determining which characterized subgroups of the circle group are (a-)factorizable, that is, can be written as the sum of two proper (a-)characterized subgroups. We correct an imprecision in one of their results, [2, Theorem 5.9] from 2017, determining the countable a-characterized subgroups of which are also a-factorizable. We also provide a revised proof of [11, Proposition 1.3] (Dikranjan, Kunen, 2007), asserting that is characterized.
近年来，Barbieri、Dikranjan、Giordano Bruno 和 Weber 在确定圆组的哪些特征子群可（a-）因式分解，即可以写成两个适当的（a-）特征子群之和的问题上取得了进展。我们纠正了他们 2017 年的一个结果[2，定理 5.9]中的不精确之处，即确定 T 的可数 a 特征化子群也是可 a 因子化的。我们还对[11，命题 1.3]（Dikranjan，Kunen，2007）进行了修订证明，断言 Q/Z 是可表征的。
{"title":"Some remarks on (a)-characterized subgroups of the circle","authors":"Nikola Bogdanovic","doi":"10.1016/j.topol.2024.109130","DOIUrl":"10.1016/j.topol.2024.109130","url":null,"abstract":"<div><div>In recent years, Barbieri, Dikranjan, Giordano Bruno and Weber have made progress on the problem of determining which characterized subgroups of the circle group are <em>(a-)factorizable</em>, that is, can be written as the sum of two proper (<em>a</em>-)characterized subgroups. We correct an imprecision in one of their results, <span><span>[2, Theorem 5.9]</span></span> from 2017, determining the countable <em>a</em>-characterized subgroups of <span><math><mi>T</mi></math></span> which are also <em>a</em>-factorizable. We also provide a revised proof of <span><span>[11, Proposition 1.3]</span></span> (Dikranjan, Kunen, 2007), asserting that <span><math><mi>Q</mi><mo>/</mo><mi>Z</mi></math></span> is characterized.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109130"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.topol.2024.109128
H.M.H. Zarenezhad, Javad Jamalzadeh
In this paper, we define a new entropy for every self-map on metric spaces, which is referred to as the Bourbaki entropy. We show, by means of an example, that the metric entropy is not necessarily equal to the Bourbaki entropy. Finally, the basic properties of the Bourbaki entropy are studied. The obtained results include the logarithmic law, invariance under conjugation, the weak addition theorem, and the completion theorem.
{"title":"A new entropy on metric spaces with respect to Bourbaki-bounded subsets","authors":"H.M.H. Zarenezhad, Javad Jamalzadeh","doi":"10.1016/j.topol.2024.109128","DOIUrl":"10.1016/j.topol.2024.109128","url":null,"abstract":"<div><div>In this paper, we define a new entropy for every self-map on metric spaces, which is referred to as the Bourbaki entropy. We show, by means of an example, that the metric entropy is not necessarily equal to the Bourbaki entropy. Finally, the basic properties of the Bourbaki entropy are studied. The obtained results include the logarithmic law, invariance under conjugation, the weak addition theorem, and the completion theorem.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109128"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.topol.2024.109127
Jorge Picado , Aleš Pultr
Open and related maps in the point-free context are studied from a consequently geometric perspective: that is, the opens are concrete well-defined subsets, images of localic maps are set-theoretic images , etc. We present a short proof of Joyal-Tierney Theorem in this setting, a (geometric) characteristic of localic maps that are just complete, and prove that open localic maps also preserve a natural type of sublocales more general than the open ones. A crucial role is played by Frobenius identities that are briefly discussed also in their general aspects.
{"title":"Frobenius identities and geometrical aspects of Joyal-Tierney Theorem","authors":"Jorge Picado , Aleš Pultr","doi":"10.1016/j.topol.2024.109127","DOIUrl":"10.1016/j.topol.2024.109127","url":null,"abstract":"<div><div>Open and related maps in the point-free context are studied from a consequently geometric perspective: that is, the opens are concrete well-defined subsets, images of localic maps are set-theoretic images <span><math><mi>f</mi><mo>[</mo><mi>U</mi><mo>]</mo></math></span>, etc. We present a short proof of Joyal-Tierney Theorem in this setting, a (geometric) characteristic of localic maps that are just complete, and prove that open localic maps also preserve a natural type of sublocales more general than the open ones. A crucial role is played by Frobenius identities that are briefly discussed also in their general aspects.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109127"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-30DOI: 10.1016/j.topol.2024.109126
Shimin Li , Jaume Llibre , Qian Tong
Poincaré compactification is very important to investigate the dynamics of vector fields in the neighborhood of the infinity, which is the main concern on the escape of particles to infinity in celestial mechanics, astrophysics, astronomy and some branches of chemistry. Since then Poincaré compactification has been extended into various cases, such as: n-dimensional polynomial vector fields, Hamiltonian vector fields, quasi-homogeneous vector fields, rational vector fields, etc.
In recent years, the piecewise smooth vector fields describing situations with discontinuities such as switching, decisions, impacts etc., have been attracted more and more attention. It is worth to notice that Poincaré compactification has been extended successfully to piecewise polynomial vector fields in 2-dimensional and 3-dimensional cases, and there are also works on n-dimensional Lipschitz continuous vector fields. The main goal of present paper is to extend the Poincaré compactification to n-dimensional piecewise polynomial vector fields which are usually discontinuous, this is a missing point in the existent literature. Thus we can investigate the dynamics near the infinity of n-dimensional piecewise polynomial vector fields. As an application we study the global phase portraits for a class of 3-dimensional piecewise linear differential systems.
庞加莱致密化对于研究无穷邻域矢量场的动力学非常重要，这也是天体力学、天体物理学、天文学和某些化学分支对粒子逸出无穷的主要关注点。从那时起，Poincaré 压缩被扩展到各种情况，如：n 维多项式矢量场、哈密顿矢量场、准均质矢量场、有理矢量场等。近年来，描述具有不连续性的情况（如切换、决策、冲击等）的片状光滑矢量场越来越受到关注。值得注意的是，Poincaré compactification 已成功地扩展到 2 维和 3 维的片断多项式向量场，也有关于 n 维 Lipschitz 连续向量场的研究。本文的主要目标是将普恩卡雷致密化扩展到 n 维的片断多项式矢量场，因为这些矢量场通常是不连续的，而这正是现有文献中缺少的一点。因此，我们可以研究 n 维片断多项式矢量场在无穷大附近的动力学。作为应用，我们研究了一类三维片断线性微分系统的全局相位肖像。
{"title":"Poincaré compactification for n-dimensional piecewise polynomial vector fields: Theory and applications","authors":"Shimin Li , Jaume Llibre , Qian Tong","doi":"10.1016/j.topol.2024.109126","DOIUrl":"10.1016/j.topol.2024.109126","url":null,"abstract":"<div><div>Poincaré compactification is very important to investigate the dynamics of vector fields in the neighborhood of the infinity, which is the main concern on the escape of particles to infinity in celestial mechanics, astrophysics, astronomy and some branches of chemistry. Since then Poincaré compactification has been extended into various cases, such as: <em>n</em>-dimensional polynomial vector fields, Hamiltonian vector fields, quasi-homogeneous vector fields, rational vector fields, etc.</div><div>In recent years, the piecewise smooth vector fields describing situations with discontinuities such as switching, decisions, impacts etc., have been attracted more and more attention. It is worth to notice that Poincaré compactification has been extended successfully to piecewise polynomial vector fields in 2-dimensional and 3-dimensional cases, and there are also works on <em>n</em>-dimensional Lipschitz continuous vector fields. The main goal of present paper is to extend the Poincaré compactification to <em>n</em>-dimensional piecewise polynomial vector fields which are usually discontinuous, this is a missing point in the existent literature. Thus we can investigate the dynamics near the infinity of <em>n</em>-dimensional piecewise polynomial vector fields. As an application we study the global phase portraits for a class of 3-dimensional piecewise linear differential systems.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109126"},"PeriodicalIF":0.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-29DOI: 10.1016/j.topol.2024.109125
Sebastián Barría
Let and denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any , the space of all unions of at most n closed intervals of is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of and are homogeneous. This partially solves an open question of A. Arhangel'skiǐ [1]. In contrast, we show that the space of closed intervals of is homogeneous.
让 A 和 S 分别表示亚历山大罗夫双箭线和索根弗雷线。我们证明，对于任意 n≥1，A 的最多 n 个封闭区间的所有联合的空间不是同质的。我们还证明了 A 和 S 的非琐收敛序列的空间是同质的。这部分解决了 A. Arhangel'skiǐ [1] 的一个未决问题。相反，我们证明了 S 的闭区间空间是同质的。
{"title":"Hyperspaces of the double arrow","authors":"Sebastián Barría","doi":"10.1016/j.topol.2024.109125","DOIUrl":"10.1016/j.topol.2024.109125","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi></math></span> and <span><math><mi>S</mi></math></span> denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, the space of all unions of at most <em>n</em> closed intervals of <span><math><mi>A</mi></math></span> is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of <span><math><mi>A</mi></math></span> and <span><math><mi>S</mi></math></span> are homogeneous. This partially solves an open question of A. Arhangel'skiǐ <span><span>[1]</span></span>. In contrast, we show that the space of closed intervals of <span><math><mi>S</mi></math></span> is homogeneous.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109125"},"PeriodicalIF":0.6,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-29DOI: 10.1016/j.topol.2024.109124
Roya Makrooni , Neda Abbasi
In this paper, we define some qualitative properties of non-autonomous discrete dynamical systems such as orbit shift continuum-wise expansivity, orbit shift persistence and orbit shift α-persistence. Then we discuss the relation between these notions and give necessary examples. Moreover, we prove that every continuum-wise expansive non-autonomous discrete system on a compact metric space is orbit shift continuum-wise expansive.
{"title":"A note on non-autonomous discrete dynamical systems","authors":"Roya Makrooni , Neda Abbasi","doi":"10.1016/j.topol.2024.109124","DOIUrl":"10.1016/j.topol.2024.109124","url":null,"abstract":"<div><div>In this paper, we define some qualitative properties of non-autonomous discrete dynamical systems such as orbit shift continuum-wise expansivity, orbit shift persistence and orbit shift <em>α</em>-persistence. Then we discuss the relation between these notions and give necessary examples. Moreover, we prove that every continuum-wise expansive non-autonomous discrete system on a compact metric space is orbit shift continuum-wise expansive.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109124"},"PeriodicalIF":0.6,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-28DOI: 10.1016/j.topol.2024.109113
Hongliang Lai , Lili Shen , Junche Yu
We investigate approach spaces generated by probabilistic metric spaces with respect to a continuous t-norm ⁎ on the unit interval . Let be the supremum of the idempotent elements of ⁎ in . It is shown that if (resp. ), then an approach space is probabilistic metrizable with respect to ⁎ if and only if it is probabilistic metrizable with respect to the minimum (resp. product) t-norm.
{"title":"On the probabilistic metrizability of approach spaces","authors":"Hongliang Lai , Lili Shen , Junche Yu","doi":"10.1016/j.topol.2024.109113","DOIUrl":"10.1016/j.topol.2024.109113","url":null,"abstract":"<div><div>We investigate approach spaces generated by probabilistic metric spaces with respect to a continuous t-norm ⁎ on the unit interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. Let <span><math><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> be the supremum of the idempotent elements of ⁎ in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. It is shown that if <span><math><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mn>1</mn></math></span> (resp. <span><math><msup><mrow><mi>k</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo><</mo><mn>1</mn></math></span>), then an approach space is probabilistic metrizable with respect to ⁎ if and only if it is probabilistic metrizable with respect to the minimum (resp. product) t-norm.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109113"},"PeriodicalIF":0.6,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-24DOI: 10.1016/j.topol.2024.109115
Tianjia Ni , Zhiying Wen
A self-similar set always possesses a self-similar topological structure coded by the shift space (symbolic space), which is considered as the coordinate system for this set. On the contrary, it is known that given a compact set K with self-similar topological structure, there may not exist a metric d such that is a self-similar set with the same topological structure. We provide an easy-to-use sufficient condition for the existence of such metric d in terms of the associated graph with respect to the self-similar topological structure. Therefore, one can easily construct a required self-similar set from the shift space by specifying the topological structure.
{"title":"Sufficient condition for a topological self-similar set to be a self-similar set","authors":"Tianjia Ni , Zhiying Wen","doi":"10.1016/j.topol.2024.109115","DOIUrl":"10.1016/j.topol.2024.109115","url":null,"abstract":"<div><div>A self-similar set always possesses a self-similar topological structure coded by the shift space (symbolic space), which is considered as the coordinate system for this set. On the contrary, it is known that given a compact set <em>K</em> with self-similar topological structure, there may not exist a metric <em>d</em> such that <span><math><mo>(</mo><mi>K</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> is a self-similar set with the same topological structure. We provide an easy-to-use sufficient condition for the existence of such metric <em>d</em> in terms of the associated graph with respect to the self-similar topological structure. Therefore, one can easily construct a required self-similar set from the shift space by specifying the topological structure.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109115"},"PeriodicalIF":0.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-23DOI: 10.1016/j.topol.2024.109114
Zhongxuan Yang, Jiajun Zhang
Given a fibred dynamical system, we introduce the notions of entropy fiber of a fibre for topological entropy, Bowen entropy and packing entropy, which quantifies the “infinitesimal change” in the dynamics of a fibre with respect to its neighboring fibres, this gives rise to an (upper semicontinuous) fibre function. Besides, we show that the topological entropy (Bowen entropy and packing entropy, resp.) of the system is the supremum of the topological entropy fiber (Bowen entropy and packing entropy, resp.) of its fibres, which provides a new perspective on the study of entropy in fibred systems.
{"title":"Local fibrations of topological entropy for fibred systems","authors":"Zhongxuan Yang, Jiajun Zhang","doi":"10.1016/j.topol.2024.109114","DOIUrl":"10.1016/j.topol.2024.109114","url":null,"abstract":"<div><div>Given a fibred dynamical system, we introduce the notions of entropy fiber of a fibre for topological entropy, Bowen entropy and packing entropy, which quantifies the “infinitesimal change” in the dynamics of a fibre with respect to its neighboring fibres, this gives rise to an (upper semicontinuous) fibre function. Besides, we show that the topological entropy (Bowen entropy and packing entropy, resp.) of the system is the supremum of the topological entropy fiber (Bowen entropy and packing entropy, resp.) of its fibres, which provides a new perspective on the study of entropy in fibred systems.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109114"},"PeriodicalIF":0.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.topol.2024.109110
Eiichi Matsuhashi
In this paper, we first provide an argument for the method used in [7] and [10] to blow up a point inside a subarc of a one-dimensional continuum to an arbitrary continuum. Next, we give an example of s Wilder continuum containing no strongly Wilder continua, no continuum-wise Wilder continua, no semiaposyndetic continua and no -continua. Also, we provide an example of a continuum such that each positive Whitney level of the hyperspace of the continuum is strongly Wilder, although the continuum itself does not contain any Wilder continua.
{"title":"Singular decomposable continua","authors":"Eiichi Matsuhashi","doi":"10.1016/j.topol.2024.109110","DOIUrl":"10.1016/j.topol.2024.109110","url":null,"abstract":"<div><div>In this paper, we first provide an argument for the method used in <span><span>[7]</span></span> and <span><span>[10]</span></span> to blow up a point inside a subarc of a one-dimensional continuum to an arbitrary continuum. Next, we give an example of s Wilder continuum containing no strongly Wilder continua, no continuum-wise Wilder continua, no semiaposyndetic continua and no <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-continua. Also, we provide an example of a continuum such that each positive Whitney level of the hyperspace of the continuum is strongly Wilder, although the continuum itself does not contain any Wilder continua.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109110"},"PeriodicalIF":0.6,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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