  The aim of this paper is to study dynamical and topological properties of a
flow in the region of influence of an isolated non-saddle set. We see, in
particular, that some topological conditions are sufficient to guarantee that
these sets are attractors or repellers. We study in detail the existence of
dissonant points of the flow, which play a key role in the description of the
region of influence of a non-saddle set. These points are responsible for much
of the dynamical and topological complexity of the system. We also study
non-saddle sets from the point of view of the Conley index theory and consider,
among other things, the case of flows on manifolds with trivial first
cohomology group. For flows on these manifolds, dynamical robustness is
equivalent to topological robustness. We carry out a particular study of
2-dimensional flows and give a topological condition which detects the
existence of dissonant points for flows on surfaces. We also prove that
isolated invariant continua of planar flows with global region of influence are
necessarily attractors or repellers.
