On contractive families and a fixed-point question of Stein
  In this paper we disprove the following conjectured generalization of the Contraction Mapping Theorem (due to J.D. Stein Jr.):   Let X be a complete metric space and let F be a finite family of self-maps of X. Suppose there is a postive constant strictly less than 1 such that, for any two points x and y of X, some member of F contracts those points by a factor of at most that constant. Then some composition of members of F has a fixed point.   We also show that the above does hold for a (continuous) commuting F containing only two maps. We conjecture that it holds for commuting F of any finite size. 
Tim D. Austin (UC, Los Angeles)