Fixed points and periodic points of orientation-reversing planar homeomorphisms

Abstract:

Two results concerning orientation-reversing homeomorphisms of the plane are proved. Let $h:\mathbb {R}^2\rightarrow \mathbb {R}^2$ 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 $\mathbb {R}^2\setminus X$ 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 $\mathbb {R}^2\setminus X$. The second result is based on a recent result of M. Bonino concerning linked periodic orbits of orientation-reversing homeomorphisms of the 2-sphere $\mathbb {S}^2$. These results generalize to orientation-reversing homeomorphisms of $\mathbb {S}^2$.