Fixed Points and Periodic Points of Orientation Reversing Planar Homeomorphisms

Abstract

Topological dynamics on surfaces is studied. The primary objects of study are orientation reversing homeomorphisms of the plane, but most of the results apply also to the orientation reversing homeomorphisms of the 2-sphere. The starting point for the present dissertation was the following problem of Krystyna Kuperberg from 1989. Suppose h is an orientation reversing homeomorphism of the plane, and there are at least n bounded components of the complement of X that are invariant under h. Must there be at least n+1 fixed points of h in X? This question is answered in the affirmative. Several other results concerning this isotopy class of homeomorphisms are proved. A separate topic of the present dissertation is constituted by periodic point theorems for plane separating circle-like continua. One of them is the following theorem. Let f be a self-map of the pseudo-circle C. Suppose that C is embedded into an annulus A, so that it separates the two components of the boundary of A. Let F be an extension of f to A (i.e. F|C=f). If F is of degree d then f has at least |d-1| fixed points. This result generalizes to all plane separating circle-like continua. In addition, several other aspects of topological dynamics on planar continua are studied relating to Sarkovskii's theorem.