In this note we prove that all
the points of a rim-compact space X at which X is not locally compact are critical
points for any local dynamical system defined on X. When a local system is global
this result is obtained by extending the global system π on X to a global system ρ on
the Freudenthal compactification Y of X, then showing that Y − X is a critical set
for ρ, and, finally, observing that Y − X contains all the points of X at which X is
not locally compact. This weaker result will appear in the author’s doctoral
dissertation and requires the use of general extension theorems proven there. For
this paper, we isolate those parts of the thesis which are pertinent to our
theorem.
