Triod twist cycles and circle rotations

TL;DR

This paper explores the relationship between cycles on a triod and circle rotations, establishing conjugacy of certain cycles via piecewise monotone maps with bounded modality.

Contribution

It introduces the concept of triod-twist cycles and proves their conjugacy to circle rotations with bounded complexity, advancing understanding of dynamical systems on triods.

Findings

Triod-twist cycles are conjugate to circle rotations.

Conjugacy is achieved via piecewise monotone maps with bounded modality.

The results connect cycle structures on triods to classical circle rotation dynamics.

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}~$\rho$, 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~$\rho$, restricted to one of its cycles.