Forcing minimal patterns of triods

TL;DR

This paper characterizes minimal patterns of triod maps that do not force other patterns with the same rotation number, showing they are conjugate to circle rotations and describing their dynamics.

Contribution

It provides a complete characterization of triod twist patterns and links them to circle rotations via piecewise monotone conjugacies.

Findings

Triod twist patterns are conjugate to circle rotations.

Complete classification of minimal triod patterns with given rotation numbers.

Dynamics of unimodal triod twist patterns are described for rational rotation numbers.

Abstract

\emph{Rotation numbers} for some maps of \emph{triods} was introduced in \cite{BMR}. The goal of this paper is to study \emph{patterns} of \emph{triods} which don't force other \emph{patterns} with the same \emph{rotation number} which we name \emph{triod twists}. We obtain their complete characterization and show that these \emph{patterns} can be conjugated to \emph{circle rotation} by a \emph{piecewise monotone} map. We also describe the dynamics of \emph{unimodal triod twist patterns} with a given rational \emph{rotation number}.