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arXiv 1012.09812010-12-05math.DS

环同胚的不变分离连续统的素端旋转数

Prime end rotation numbers of invariant separating contunua of annular homeomorphisms

Shigenori Matsumoto

备注：: 7 pages
发表信息：: Proc. A. M. S. 140(2012), 839-845

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摘要：

令$f$是与恒等式同位素的闭环$A$的同胚，令$X\subset {\rm Int}A$是将$A$分成两个域的$f$不变连续体，即上域$U_+$和下域$U_-$ .将$f$的提升固定到$A$的泛覆盖上，利用$X$上的不变概率定义了$X$的旋转集$\tilde ρ(X)$，以及$U_\pm$的素端旋转数$\checkρ_\pm$.本文的目的是证明对于任何分离不变连续统$X$，$\checkρ_\pm$都属于$\tildeρ(X)$.

英文摘要：

Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$, as well as the prime end rotation number $\checkρ_\pm$ of $U_\pm$. The purpose of this paper is to show that $\checkρ_\pm$ belongs to $\tildeρ(X)$ for any separating invariant continuum $X$.

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