arXiv math/04064422004-06-22math.DS

Continuum-wise expansive homeomorphisms on Peano continua

Jana Rodriguez Hertz

英文摘要：

On a Peano continuum, all local stable and unstable components of a continuum-wise expansive homeomorphism are non trivial. In particular, there is sensitive dependence on initial conditions. This generalizes results in \cite{h,l} about lack of Lyapunov stable points (weak sinks) and existence of non trivial stable and unstable components for expansive homeomorphisms on Peano continua. We also use this fact to generalize a result in \cite{ktt}: a Peano curve $X$ admitting a continuum-wise expansive homeomorphism is nowhere rim-countable. However, it is not resolved the question of whether such a dynamics could be possible in a locally planar Peano curve. Some other questions are posed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Discrete and Continuous Topological Dynamics: Fields of Cross Sections and Expansive Flows

In this article we consider the general problem of translating definitions and results from the category of discrete-time dynamical systems to the category of flows. We consider the dynamics of homeomorphisms and flows on compact metric spaces, in particular Peano continua. As a translating tool, we construct continuous, symmetric and monotonous fields of local cross sections for an arbitrary flow without singular points. Next, we use this structure in the study of expansive flows on Peano continua. We show that expansive flows admit no stable point and that every point contains a non-trivial …

2015-02-09math.DS

Peano连续统上的混沌

Chaos on Peano continua

本研究推广Agronsky和Ceder的结果，证明Peano连续统存在精确Devaney混沌的连续变换，界定含一维连续统、绝对邻域收缩核的Peano连续统类，通过局部小扰动证明该类中混沌系统的稠密性与相关通有性结论。

2025-09-01math.DS

Lyapunov exponents for expansive homeomorphisms

Let (M,d) be a compact metric space and f:M --> M an expansive homeomorphism. We define Lyapunov exponents L(f,m)_{max} and l(f,mu)_{min} for an f-invariant measure m. When L(f,m)_{max} > 0 and l(f,mu)_{min} < 0 can be interpreted as a weak form of hyperbolicity for f. We prove that if M is a Peano space then there is g>0 such that L(f,m)_{max} > g and l(f,m)_{min}< - g. We also show that the hypothesis that M is a Peano space is necessary to obtain the maximal Lyapunov exponent positive and the minimal Lyapunov exponent negative. Moreover we define Lyapunov exponents for K, a compact f-invari…

2017-04-18math.DS

cw-扩张流的稳定/不稳定连续体

Stable/unstable continua of cw-expansive flows

本文提出三种不同定义的局部稳定/不稳定集，并证明cw-扩张流中存在直径一致的连续体。

2024-10-01math.DS

Sensitive open map semigroups on Peano continua having a free arc

Let $X$ be a Peano continuum having a free arc and let $C^0(X)$ be the semigroup of continuous self-maps of $X$. A subsemigroup $F\subset C^0(X)$ is said to be sensitive, if there is some constant $c>0$ such that for any nonempty open set $U\subset X$, there is some $f\in F$ such that the diameter ${\rm diam}(f(U))>c$. We show that if $X$ admits a sensitive commutative subsemigroup $F$ of $C^0(X)$ consisting of continuous open maps, then either $X$ is an arc, or $X$ is a circle.

2018-04-07math.DS

The nonexistence of expansive actions on Suslinian continua by groups of subexponential growth

We show that if $G$ is a finitely generated group of subexponential growth and $X$ is a Suslinian continuum, then any action of $G$ on $X$ cannot be expansive.

2021-04-13math.DS

更多相关论文收起更多论文

Peano continua with self regenerating fractals

We deal with the question of Masayoshi Hata: is every Peano continuum a topological fractal? A compact space $X$ is a topological fractal if there exists $\mathcal{F}$ a finite family of self-maps on $X$ such that $X=\bigcup_{f\in\mathcal{F}}f(X)$ and for every open cover $\mathcal{U}$ of $X$ there is $n\in\mathbb{N}$ such that for all maps $f_1,\dots,f_n\in\mathcal{F}$ the set $f_1\circ\dots\circ f_n(X)$ is contained in some set $U\in\mathcal{U}$. In the paper we present some idea how to extend a topological fractal and we show that a Peano continuum is a topological fractal if it contains so…

2021-06-21math.DSmath.GN

A Core Decomposition of Compact Sets in the Plane

A Peano continuum means a locally connected continuum. A compact metric space is called a \emph{Peano compactum} if all its components are Peano continua and if for any constant $C>0$ all but finitely many of its components are of diameter less than $C$. Given a compact set $K\subset\mathbb{C}$, there usually exist several upper semi-continuous decompositions of $K$ into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the \emph{core decomposition of $K$ with P…

2018-11-21math.DS

曲面与其他不存在 Generic Chains 的 Peano 连续统

Surfaces and other Peano Continua with no Generic Chains

本文证明了一大类拓扑空间（包括维数≥3的流形、多数紧致曲面及齐性Peano连续统）不存在generic链，并在强局部齐性空间中证明链无法被可数结构分类且同胚群具有不可度化泛极小流。

2025-02-02math.DSmath.GNmath.LO

首次敏感同胚

First-time sensitive homeomorphisms

本文提出同胚的首次敏感性概念，证明其涵盖连续统式膨胀同胚、Hilbert方体移位映射及部分部分双曲微分同胚，并在附加技术假设下推出正拓扑熵。

2023-04-06math.DS

Topological entropy and IE-tuples of indecomposable continua

In this paper, we define a new notion of "freely tracing property by free chains" on $G$-like continua and we prove that a positive topological entropy homeomorphism on a $G$-like continuum admits a Cantor set $Z$ such that every tuple of finite points in $Z$ is an $IE$-tuple of $f$ and $Z$ has the freely tracing property by free chains. Also, by use of this notion, we prove the following theorem: If $G$ is any graph and a homeomorphism $f$ on a $G$-like continuum $X$ has positive topological entropy, then there is a Cantor set $Z$ which is related to both the chaotic behaviors of Kerr and Li …

2017-03-17math.DS

曲面上的连续统双曲同胚

Continuum-wise hyperbolic homeomorphisms on surfaces

本文研究曲面连续统双曲同胚的动力学，证明含有限多个脊的 $cw_F$-双曲同胚为 $cw_2$-双曲，并证明 $cw_3$-双曲性蕴含 $cw_2$-双曲性，同时揭示脊的孤立性。

2023-05-15math.DS