Twist like behavior in non-twist patterns of triods

TL;DR

This paper investigates special patterns on triods that exhibit twist-like behavior in their rotation numbers, revealing new phenomena not present in circle maps and providing methods to construct such patterns.

Contribution

It introduces and characterizes 'strangely ordered' patterns on triods, which mimic twist patterns without being actual twists, and offers an algorithm to construct these patterns with arbitrary rotation pairs.

Findings

Existence of 'strangely ordered' patterns with rotation numbers at endpoints of their forced rotation intervals.

These patterns are semi-conjugate to circle rotations via piecewise monotone maps.

New phenomena in rotation theory for triods compared to circle maps.

Abstract

We prove a sufficient condition for a \emph{pattern} $\pi$ on a \emph{triod} $T$ to have \emph{rotation number} $\rho_{\pi}$ coincide with an end-point of its \emph{forced rotation interval} $I_{\pi}$. Then, we demonstrate the existence of peculiar \emph{patterns} on \emph{triods} that are neither \emph{triod twists} nor possess a \emph{block structure} over a \emph{triod twist pattern}, but their \emph{rotation numbers} are an end point of their respective \emph{forced rotation intervals}, mimicking the behavior of \emph{triod twist patterns}. These \emph{patterns}, absent in circle maps (see \cite{almBB}), highlight a key difference between the rotation theories for \emph{triods} (introduced in \cite{BMR}) and that of circle maps. We name these \emph{patterns}: ``\emph{strangely ordered}" and show that they are semi-conjugate to circle rotations via a piece-wise monotone map. We conclude by providing an algorithm to construct unimodal \emph{strangely ordered patterns} with arbitrary \emph{rotation pairs}.