Topological entropy of transitive maps of a tree
by
Xiandong Ye
University of Science and Technology of China

Let T be a tree, End(T) be the number of ends of T, and L(T) be the infimum of topological entropies of transitive maps of T. We give a simple proof of the estimate L(T) >= [ log2/End(T)]. We also divide the set of all trees into pairwise disjoint subsets, P(i), i in {0} \cup \N, and prove that L(T)=[ log2/(End(T)-i)] if T in P(i) with i=0, 1, L(T) <= [ log2/(End(T)-i)] if T in P(i) with i > 1 and infinite subsets P'(i) subset or equal P(i), for which L(T)=[ log2/(End(T)-i)] if T in 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. (Topology, 1997). Moreover, we discuss the above problem for some class of trees.

Date received: August 20, 1999