Jeroen S.W. Lamb & Matthew Nicol:

On symmetric omega-limit sets in reversible flows

Abstract:

Let $\Gamma \subset O(n)$ be a finite  group acting orthogonally
on $\Real^n$. We say that $\Gamma$ is a reversing symmetry group
of the  flow $f^t$ 
if $\Gamma$ has an  index
two subgroup $\tilde{\Gamma}$ whose elements commute with $f^t$ 
and for all elements $\rho \in \Gamma - \tilde{\Gamma}$ and all $t$,
$f^t \circ\rho (x) = \rho\circ f^{-t} (x)$ .
In dimensions $n=1,2$ we describe all symmetry
groups of $\omega$-limit sets for such reversible flows.
In case $n\ge 1$ we  give group and representation theoretic restrictions
on possible symmetry groups and show that for subgroups of $\tilde{\Gamma}$
our conditions are necessary and sufficient. We also describe in detail
the possible symmetries of periodic orbits. 

Finally, we show that if a Liapunov stable $\omega$-limit set is fixed setwise
by a reversing symmetry then it is transitive.