FIXED POINTS AND PERIODIC POINTS OF ORIENTATION-REVERSING PLANAR HOMEOMORPHISMS

Two results concerning orientation-reversing homeomorphisms of the plane are proved. Let h : R(2) -> R(2) be an orientation-reversing planar homeomorphism with a continuum X invariant (i.e. h(X) = X). First, suppose there are at least n bounded components of R(2) \ X that are invariant under h. Then there are at least n 1 components of the fixed point set of h in X. This provides an affirmative answer to a question posed by K. Kuperberg. Second, suppose there is a k-periodic orbit in X with k > 2. Then there is a 2-periodic orbit in X, or there is a 2-periodic component of R(2) \ X. The second result is based on a recent result of M. Bonino concerning linked periodic orbits of orientation-reversing homeomorphisms of the 2-sphere S(2). These results generalize to orientation-reversing homeomorphisms of S(2).