DYNAMICS OF HOMEOMORPHISMS OF REGULAR CURVES

Let f : X -> X be a homeomorphism of a regular curve X. We prove that the space of minimal sets of f is closed in the hyperspace 2(X) of closed subsets of X endowed with the Hausdorff metric. As a consequence, we establish the equivalence between pointwise periodicity of f and the Hausdorffness of the orbit space X/f. Moreover, we prove that the nonwandering set Omega(f) is equal to the set of recurrent points of f and we study the continuity of the map omega(f) : X -> X-2, x bar right arrow omega(f)(x). We show for instance the equivalence between the continuity of omega(f) and the equality between the omega-limit set and the omega-limit set of every point in X. Finally, we prove that there is only one (infinite) minimal set when there is no periodic point.