Topological entropy of transitive maps of a tree

摘要

Let T be a tree, End(T) be the number of ends of T and let L(T) be the infimum of topological entropies of transitive maps of T. We give an elementary approach to the estimate that L(T) ≥ (1/End(T))log2. We also divide the set of all trees (up to homeomorphisms) into pairwise disjoint subsets P(i), i ∈{0} ∪ N and prove that L(T) = (1/End(T) - i)) log2 if T ∈P(i) with i = 0, 1, and L(T) ≤ (respectively = ) (1/End(T) - i)) log2 if T ∈P(i) (respectively T ∈P'(i) with i ≥ 2, where P'(i) is an infinite subset of P(i). Furthermore, we show that there is a tree T such that the topological entropy of each transitive map of T is larger than L(T), and hence disprove a conjecture of Alseda et al(1997).