FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES

We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map f has positive entropy if and only if some iterate f^k has a periodic orbit with three aligned points consecutive in time, that is, a triplet (a, b, c) such that f^k(a) = b, f^k(b) = c and b belongs to the interior of the unique interval connecting a and c (a forward triplet of f^k). We also prove a new criterion of entropy zero for simplicial n-periodic patterns P based on the non existence of forward triplets of f^k for any 1 ≤ k < n inside P. Finally, we study the set Xn of all n-periodic patterns P that have a forward triplet inside P. For any n, we define a pattern that attains the minimum entropy in Xn and prove that this entropy is the unique real root in (1, ∞) of the polynomial xⁿ − 2x − 1.