5 research outputs found
On the dynamics of sup-norm non-expansive maps
We present several results for the periods of periodic points of sup-norm non-expansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map , where , is at most . This upper bound is smaller than 3n and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is . We give a proof of this conjecture. To obtain the results we use combinatorial and geometric arguments. In particular, we analyse the cardinality of anti-chains in certain partially ordered sets
Spectral Theorem for Convex Monotone Homogeneous Maps, and Ergodic Control
We consider convex maps f:R^n -> R^n that are monotone (i.e., that preserve
the product ordering of R^n), 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
non-empty, is isomorphic to a convex inf-subsemilattice of 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 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.Comment: 38 pages, 13 Postscript figure
Periods of nonexpansive operators on finite L1-spaces
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