The Positive Entropy Kernel for Some Families of Trees

The fact that a continuous self-map of a tree has positive topological entropy is related to the amount of different gods (greatest odd divisors) exhibited by its set of periods. Llibre & Misiurewicz [11] and Blokh [9] give generic upper bounds for the maximum number of gods that a zero entropy tree map f : T −→ T can exhibit, in terms of the number of endpoints and edges of T . In this paper we compute exactly the minimum of the positive integers n such that the entropy of each tree map f : T −→ T exhibiting more than n gods is necessarily positive, for the family of trees which have a subinterval containing all the branching points (this family includes the interval and the stars). We also compute which gods are admissible for such maps.