Message from Prof. Roger Nussbaum about the problem
described to me by him and Bas Lemmens 

Dear Doron,

"Omega limit sets of nonexpansive maps: 
finiteness and cardinality estimates", Differential and Integral 
Equations 3 (1990), pages 523-540. See equation (12), page 525. of that 
article. The cases n=1 and n=2 are known and easy. Richard Lyons and I 
proved the case n=3  in "On transitive and commutative finite groups of 
isometries" in Fixed Point Theory and Applications, K.-K. Tan, editor, 
World Scientific, Singapore, New Jersey, London, Hong Kong, 1992, pages 
189-228.  The proof for n=3 involves a laborious case-by-case analysis. 
In the paper "On the dynamics of sup norm nonexpansive maps" (Ergodic 
Theory and Dynamical Systems 25 (2005), 61-71), Bas Lemmens and Michael 
Scheutzow proved that the conjecture is true if 2^n is replaced by 
max{c(k,n)2^k : 0<k<=n}, where c(k,n):="n, choose k", the kth binomial 
coefficient. This is the best known general result, though Richard Lyons 
and I have proved that if p is a prime larger than 2^n, then p cannot be 
the period of a periodic point of a sup norm nonexpansive map of R^n to 
R^n.  The 2^n conjecture remains open for n=4.