Nielsen classes of orientation reversing homeomorphisms and complementary domains of plane separating invariant continua

Let h be an orientation reversing planar homeomorphism and X be an invariant plane separating continuum. We show that there is a natural linear order on the invariant components of R 2 ∖ X that resemble the one found in connected unions of circles invariant under the reflection r ( x , y ) = ( − x , y ) . The main result relates to the Nielsen fixed point theory and work of Krystyna Kuperberg on fixed points of planar homeomorphisms in invariant continua.