Spectral theorem for convex monotone homogeneous maps, and ergodic control
By Marianne Akian, Stephane Gaubert. 2001
65 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/13f011af0d34631b336e12d16744ec147a0ff43b
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Citations in the corpus
- The max-plus Martin boundary (2004, unknown): “The reader may consult [MS92, BCOQ92, Bap98, GM02, AG03 , AGW04] for more background on max-plus spectral theory.”
- The Perron-Frobenius theorem for homogeneous, monotone functions (2001, methodology): “Functions which are homogeneous and monotone on the positive cone provide a natural generalisation of nonnegative matrices and have a correspondingly wide range of application: in population biology, [30], mathematical economics, [11, 27], discrete event systems, [3, 17], idempotent analysis, [16, 2”