449 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
A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space
A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik
(independently from each other) to metric spaces leads to a stronger version of
ultrahomogeneity of the infinite random graph , the universal Urysohn metric
space \Ur, and other related objects. We show how the result can be used to
average out uniform and coarse embeddings of \Ur (and its various
counterparts) into normed spaces. Sometimes this leads to new embeddings of the
same kind that are metric transforms and besides extend to affine
representations of various isometry groups. As an application of this
technique, we show that \Ur admits neither a uniform nor a coarse embedding
into a uniformly convex Banach space.Comment: 23 pages, LaTeX 2e with Elsevier macros, a significant revision
taking into account anonymous referee's comments, with the proof of the main
result simplified and another long proof moved to the appendi
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