Abstract
Let X be a local dendrite with countably many endpoints and f:X→X be a continuous map without periodic points. We prove that the topological entropy of f is zero; f has a unique minimal subset; the closure of the set of recurrent points is contained in the minimal subcontinuum containing all simple closed curves. We show by an example that the above conclusions do not hold if the countability assumption on the endpoint set is removed. Some new phenomena different from graph maps are noticed.