Prime end rotation numbers of invariant separating continua of annular homeomorphisms

Matsumoto, Shigenori

doi:10.1090/S0002-9939-2011-11435-7

Prime end rotation numbers of invariant separating continua of annular homeomorphisms
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Proc. Amer. Math. Soc. 140 (2012), 839-845 Request permission

Abstract:

Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset \textrm {Int}A$ 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 $\tilde \rho (X)$ of $X$ by means of the invariant probabilities on $X$, as well as the prime end rotation number $\check \rho _\pm$ of $U_\pm$. The purpose of this paper is to show that $\check \rho _\pm$ belongs to $\tilde \rho (X)$ for any separating invariant continuum $X$.

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