International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 19, Pages 3025-3033
The minimum tree for a given zero-entropy period
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 , which is the minimum number of endpoints of a tree admitting a zero-entropy map with a periodic orbit of period ?
We prove that , where is the decomposition of into a product of primes such that for . As a corollary, we get a criterion to decide whether a map defined on a tree with endpoints has positive entropy: if has a periodic orbit of period with , then the topological entropy of is positive.