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Abstract

The goal of this paper is to study cycles on \emph{triods} whose dynamics emulate those of a \emph{mixing map}, referred to as \emph{mixing cycles}. We establish a necessary and sufficient condition for a cycle on a \emph{triod} $Y$ to be \emph{mixing}. We further show that the rule governing the co-existence of periods of \emph{mixing} cycles of a map on a \emph{triod} induces a linear ordering of the natural numbers, which remains stable under small perturbations of the map. Finally, we demonstrate the existence of exceptional \emph{mixing} cycles on \emph{triods} whose \emph{rotation numbers} coincide with an endpoint of the corresponding \emph{forced rotation interval}.