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

Abstract

Let $X$ be a dendroid, $f:X\to X$ 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.