On homoclinic points and topological entropy of continuous maps on one - dimensional ramified continua

Let X be a dendroid, f:X→ X be a continuous map, p be a periodic point of f and let x be a homoclinic point in X to the periodic point p. We study the properties of the homoclinic point x and the unstable manifold of the point p. We investigate the local structure of X under which the existence of a homoclinic point implies the positive topological entropy of f. We also present differences in the properties of homoclinic points and the unstable manifolds of periodic points for continuous maps defined on dendroids, dendrites and finite trees.