On the dynamics of sup-norm nonexpansive maps

We present several results for the periods of periodic points of sup-norm nonexpansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map f : M → M , where M ⊂ R, is at most maxk 2 ( n k ) . This upper bound is smaller than 3 and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions f : R → R Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is ( n n/2 ). 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.