Fixed points and periodic points of orientation-reversing planar homeomorphisms

Abstract

Two results concerning orientation-reversing homeomorphisms of the plane are proved. Let h : R 2 → R 2 h:\mathbb {R}^2\rightarrow \mathbb {R}^2 be an orientation-reversing planar homeomorphism with a continuum X X invariant (i.e. h ( X ) = X h(X)=X ). First, suppose there are at least n n bounded components of R 2 ∖ X \mathbb {R}^2\setminus X that are invariant under h h . Then there are at least n + 1 n+1 components of the fixed point set of h h in X X . This provides an affirmative answer to a question posed by K. Kuperberg. Second, suppose there is a k k -periodic orbit in X X with k > 2 k>2 . Then there is a 2-periodic orbit in X X , or there is a 2-periodic component of R 2 ∖ X \mathbb {R}^2\setminus 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 \mathbb {S}^2 . These results generalize to orientation-reversing homeomorphisms of S 2 \mathbb {S}^2 .