FIXED-POINTS OF PLANE CONTINUA

Suppose M is a plane continuum, D is a decomposition of M, and each element of D is a uniquely arcwise connected set. Our principal theorem states that every map of M that preserves the elements of D has a fixed point. It follows that every arc-component-preserving map of a plane continuum that does not contain a simple closed curve has a fixed point. This result generalizes the author's theorem [17] that every uniquely arcwise connected plane continuum has the fixed-point property. Our principal theorem also applies to planar dynamical systems. Suppose psi is a continuous flow on the plane. Suppose M is an invariant continuum under psi and D is the collection of orbits of psi in M. Then, according to our principal theorem, some element of D is a point or a simple closed curve. Hence, every invariant continuum under psi contains an equilibrium point or a closed orbit. This result implies the Poincare-Bendixson theorem, a compact limit set of an orbit of psi that does not contain an equilibrium point is a closed orbit.