  Let K denote a compact invariant set for a strongly monotone semiflow in an
ordered Banach space E, satisfying standard smoothness and compactness
assumptions. Suppose the semiflow restricted to K is chain transitive. The main
result is that either K is unordered, or else K is contained in totally
ordered, compact arc of stationary points; and the latter cannot occur if the
semiflow is real analytic and dissipative. As an application, entropy is 0 when
E = R^3 . Analogous results are proved for maps. The main tools are results of
Mierczynski [27 ] and Terescak [37 ]
