Orientation preserving homeomorphisms of the plane having BP-chain recurrent points

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-05-20 DOI:10.1016/j.topol.2025.109427

Jiehua Mai , Kesong Yan , Fanping Zeng

引用次数: 0

Abstract

More than a century ago, L. E. J. Brouwer proved a famous theorem, which says that any orientation preserving homeomorphism of the plane having a periodic point must have a fixed point. In recent years, there are still some authors giving various proofs of this fixed point theorem. In [7], Fathi showed that the condition “having a periodic point” in this theorem can be weakened to “having a non-wandering point”. In this paper, we first give a new proof of Brouwer's theorem, which is relatively simpler and the statement is more compact. Further, we propose a notion of BP-chain recurrent points, which is a generalization of the concept of non-wandering points, and we prove that if an orientation preserving homeomorphism of the plane has a BP-chain recurrent point, then it has a fixed point. This further weakens the condition in the Brouwer's fixed point theorem on plane.

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具有bp链循环点的平面的保取向同胚

一个多世纪以前，L. E. J.布劳威尔证明了一个著名的定理，即任何具有周期点的平面的方向保持同胚，必须有一个不动点。近年来，仍有一些作者给出了这个不动点定理的各种证明。在[7]中，Fathi证明了该定理中“有周期点”的条件可以弱化为“有非游荡点”。在本文中，我们首先给出了布劳威尔定理的一个新的证明，它相对来说更简单，表述也更紧凑。进一步，我们提出了bp -链循环点的概念，这是对非游荡点概念的推广，并证明了如果平面的保向同胚存在bp -链循环点，那么它就存在不动点。这进一步削弱了平面上布劳威尔不动点定理的条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.