  A continuous action of a group G on a compact metric space has sensitive
dependence on initial conditions if there is a number e>0 such that for any
open set U we can find g in G such that g.U has diameter greater than e. We
prove that if a G action preserves a probability measure of full support, then
the system is either minimal and equicontinuous, or has sensitive dependence on
initial conditions. This generalizes the invertible case of a theorem of
Glasner and Weiss. We prove that when a finitely generated, solvable group,
acts and certain cyclic subactions have dense sets of minimal points, then the
system has sensitive dependence on initial conditions. Additionally, we show
how to construct examples of non-compact monothetic groups, and transitive,
non-minimal, almost-equicontinuous, recurrent, group actions.
