One-Dimensional Systems Without Periodic Points

If X is a local dendrite and f : X -> X is a continuous map without periodic points then we prove that the canonical graph (the minimal subcontinuum containing all simple closed curves) contains one and only one minimal subset and that this minimal subset is circumferential. Furthermore, we show the rigidity of the dynamics of f when the set of endpoints of X is closed and countable. However, if we relax either the closedness or countability condition on the set of endpoints then several types of chaos may appear which shows the richness of the dynamics comparing with that of a graph map without periodic points.