Positive circuits and maximal number of fixed points in discrete dynamical systems

Details Positive circuits and maximal number of fixed points in discrete dynamical systems Positive circuits and maximal number of fixed points in discrete dynamical systems

2008-07-26

arxiv.org/abs/0807.4229v3

Computer Science

Discrete Mathematics

13 pages

Scientific paper

We consider the Cartesian product X of n finite intervals of integers and a map F from X to itself. As main result, we establish an upper bound on the number of fixed points for F which only depends on X and on the topology of the positive circuits of the interaction graph associated with F. The proof uses and strongly generalizes a theorem of Richard and Comet which corresponds to a discrete version of the Thomas' conjecture: if the interaction graph associated with F has no positive circuit, then F has at most one fixed point. The obtained upper bound on the number of fixed points also strongly generalizes the one established by Aracena et al for a particular class of Boolean networks.

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