International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 19, Pages 3025-3033

The minimum tree for a given zero-entropy period

Esther Barrabés and David Juher

Departament d'Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, Girona 17071, Spain

Received 5 May 2005; Revised 22 September 2005

Copyright © 2005 Esther Barrabés and David Juher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We answer the following question: given any n, which is the minimum number of endpoints en of a tree admitting a zero-entropy map f with a periodic orbit of period n? We prove that en=s1s2ski=2ksisi+1sk, where n=s1s2sk is the decomposition of n into a product of primes such that sisi+1 for 1i<k. As a corollary, we get a criterion to decide whether a map f defined on a tree with e endpoints has positive entropy: if f has a periodic orbit of period m with em>e, then the topological entropy of f is positive.