Negative circuits and sustained oscillations in asynchronous automata networks

The biologist Ren\'e Thomas conjectured, twenty years ago, that the presence of a negative feedback circuit in the interaction graph of a dynamical system is a necessary condition for this system to produce sustained oscillations. In this paper, we state and prove this conjecture for asynchronous automata networks, a class of discrete dynamical systems extensively used to model the behaviors of gene networks. As a corollary, we obtain the following fixed point theorem: given a product $X$ of $n$ finite intervals of integers, and a map $F$ from $X$ to itself, if the interaction graph associated with $F$ has no negative circuit, then $F$ has at least one fixed point.