Cycle times and Xed Points of Min-max Functions

The dynamic behaviour of discrete event systems with only maximum timing constraints (or, dually, only minimum timing constraints) can be studied by linear methods based on max-plus algebra, as discussed in the recent book \Synchronization and Linearity" by Bac-celli, Cohen, Quadrat and Olsder. Systems with both maximum and minimum constraints are non-linear from this perspective and only limited results about their behaviour have so far been obtained In this paper we describe some new results and conjectures about such systems which generalize the initial earlier work and shed a new light on aspects of the linear theory. Our main result is a necessary and suucient condition for the existence of a xed point (eigenvector) for any min-max function.