Periods of nonexpansive operators on finite l1-spaces
European Journal of Combinatorics
Almost Sure Stabilizability and Riccati's Equation of Linear Systems with Random Parameters
SIAM Journal on Control and Optimization
Competitive Markov decision processes
Competitive Markov decision processes
Optimization over Time
Some Ergodic Results on Stochastic Iterative Discrete Events Systems
Discrete Event Dynamic Systems
Methods and Applications of (MAX, +) Linear Algebra
STACS '97 Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science
From max-plus algebra to nonexpansive mappings: a nonlinear theory for discrete event systems
Theoretical Computer Science
Characterisation of ergodic upper transition operators
International Journal of Approximate Reasoning
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We consider convex maps f : Rn → Rn that are monotone (i.e., that preserve the product ordering of Rn), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex inf-subsemilattice of Rn, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.