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Poincaré compactification for n-dimensional piecewise polynomial vector fields: Theory and applications n 维片断多项式矢量场的庞加莱压缩：理论与应用
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-30 DOI: 10.1016/j.topol.2024.109126
Shimin Li , Jaume Llibre , Qian Tong
{"title":"Poincaré compactification for n-dimensional piecewise polynomial vector fields: Theory and applications","authors":"Shimin Li ,&nbsp;Jaume Llibre ,&nbsp;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}
引用次数: 0
Hyperspaces of the double arrow 双箭头的超空间
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-29 DOI: 10.1016/j.topol.2024.109125
Sebastián Barría
{"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}
引用次数: 0
A note on non-autonomous discrete dynamical systems 关于非自治离散动力系统的说明
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-29 DOI: 10.1016/j.topol.2024.109124
Roya Makrooni , Neda Abbasi
{"title":"A note on non-autonomous discrete dynamical systems","authors":"Roya Makrooni ,&nbsp;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}
引用次数: 0
On the probabilistic metrizability of approach spaces 论方法空间的概率元可操作性
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-28 DOI: 10.1016/j.topol.2024.109113
Hongliang Lai , Lili Shen , Junche Yu
{"title":"On the probabilistic metrizability of approach spaces","authors":"Hongliang Lai ,&nbsp;Lili Shen ,&nbsp;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>&lt;</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}
引用次数: 0
Sufficient condition for a topological self-similar set to be a self-similar set 拓扑自相似集合是自相似集合的充分条件
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-24 DOI: 10.1016/j.topol.2024.109115
Tianjia Ni , Zhiying Wen
{"title":"Sufficient condition for a topological self-similar set to be a self-similar set","authors":"Tianjia Ni ,&nbsp;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}
引用次数: 0
Local fibrations of topological entropy for fibred systems 纤维系统拓扑熵的局部纤维化
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-23 DOI: 10.1016/j.topol.2024.109114
Zhongxuan Yang, Jiajun Zhang
{"title":"Local fibrations of topological entropy for fibred systems","authors":"Zhongxuan Yang,&nbsp;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}
引用次数: 0
Singular decomposable continua 奇异可分解连续
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-21 DOI: 10.1016/j.topol.2024.109110
Eiichi Matsuhashi
{"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}
引用次数: 0
Egorov ideals 叶戈罗夫的理想
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-21 DOI: 10.1016/j.topol.2024.109112
Adam Kwela
{"title":"Egorov ideals","authors":"Adam Kwela","doi":"10.1016/j.topol.2024.109112","DOIUrl":"10.1016/j.topol.2024.109112","url":null,"abstract":"<div><div>We study Egorov ideals, that is ideals on <em>ω</em> for which the Egorov's theorem for ideal versions of pointwise and uniform convergences holds. We show that a non-pathological <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> ideal is Egorov if and only if it is countably generated. In particular, up to isomorphism, there are only three non-pathological <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> Egorov ideals. On the other hand, we construct <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msup></math></span> pairwise non-isomorphic Borel Egorov ideals. Moreover, we characterize when a product of ideals is Egorov.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109112"},"PeriodicalIF":0.6,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530944","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}
引用次数: 0
Various notions of shadowing in triangular system and its component systems 三角形系统及其组成系统中的各种阴影概念
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-18 DOI: 10.1016/j.topol.2024.109109
Deepanshu Dhawan, Puneet Sharma
{"title":"Various notions of shadowing in triangular system and its component systems","authors":"Deepanshu Dhawan,&nbsp;Puneet Sharma","doi":"10.1016/j.topol.2024.109109","DOIUrl":"10.1016/j.topol.2024.109109","url":null,"abstract":"<div><div>In this article, we investigate various forms of shadowing for a general triangular system. In particular, we relate various notions of shadowing for a triangular system with various notions of shadowing in the component systems. We prove that if the base <em>f</em> for <em>T</em> is transitive then shadowing in the base map and the non-autonomous system generated by a transitive point ensures shadowing of the triangular system. We prove that if the base map for <em>T</em> is expansive then shadowing in the triangular system ensures shadowing in the component systems. We prove that if the non-autonomous component systems form a synchronized family and the base map possesses a globally attracting fixed point <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> then eventual shadowing in system generated by <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> ensures eventual shadowing in each of the non-autonomous component systems. We also investigate chain transitivity and chain mixing for a general triangular system.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109109"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534568","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}
引用次数: 0
Non-abelian tensor product and circular orderability of groups 非阿贝尔张量积与群的循环有序性
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-18 DOI: 10.1016/j.topol.2024.109111
Maxim Ivanov
{"title":"Non-abelian tensor product and circular orderability of groups","authors":"Maxim Ivanov","doi":"10.1016/j.topol.2024.109111","DOIUrl":"10.1016/j.topol.2024.109111","url":null,"abstract":"<div><div>For a group <em>G</em> we consider its tensor square <span><math><mi>G</mi><mo>⊗</mo><mi>G</mi></math></span> and exterior square <span><math><mi>G</mi><mo>∧</mo><mi>G</mi></math></span>. We prove that for a circularly orderable group <em>G</em>, under some assumptions on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, its exterior square and tensor square are left-orderable. This yields an obstruction for a circularly orderable group <em>G</em> to have torsion. We apply these results to study circular orderability of tabulated virtual knot groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109111"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552383","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}
引用次数: 0
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