Triod twist cycles and circle rotations

Abstract

We study the problem of relating cycles on a \emph{triod} $Y$ to \emph{circle rotations}. We prove that the simplest cycles on a \emph{triod}~$Y$ with a given \emph{rotation number}~$\ddot{I}$, called \emph{triod--twist cycles} are conjugate, via a piece-wise monotone map of \emph{modality} at most~$m+3$, where~$m$ is the \emph{modality} of~$P$ to the rotation on~$S^1$ by angle~$\ddot{I}$, restricted to one of its cycles.

20 pages, 10 figures