  We give a new lower bound on the number of connected components of the space
of representations of a surface group into the group of orientation preserving
homeomorphisms of the circle. Precisely, for the fundamental group of a genus g
surface, we show there are at least k^(2g) + 1 connected components containing
representations with Euler number (2g-2)/(k). We also show that certain
representations are rigid, meaning that all deformations lie in the same
semiconjugacy class. Our methods apply to representations of surface groups
into finite covers of PSL(2,R) and into Diff+(S^1) as well, in which case we
recover theorems of W. Goldman and J. Bowden.
  The key technique is an investigation of local maximality phenomena for
rotation numbers of products of circle homeomorphisms, using techniques of
Calegari-Walker. This is a new approach to studying deformation classes of
group actions on the circle, and may be of independent interest.
