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}.
Categories: math.DS