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Spectral theorem for convex monotone homogeneous maps, and ergodic control. (English) Zbl 1030.47048
In this paper, the authors consider a convex map $$f:\mathbb R^n\to\mathbb R^n$$ which is monotone and nonexpansive for the sup-norm and show that the fixed point set of $$f$$, when it is nonempty, is isomorphic to a convex inf-subsemilattice of $$\mathbb R^n$$, 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 a uniqueness result for the bias vector of ergodic control problems, which generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen.

##### MSC:
 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 93E20 Optimal stochastic control 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 15A80 Max-plus and related algebras 15B48 Positive matrices and their generalizations; cones of matrices
##### Keywords:
Nonexpansive maps; periodic orbits; eigenspace; spectral theorem
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