PERIODS IMPLYING ALMOST ALL PERIODS FOR TREE MAPS

Let X be a compact tree, f : X --> X be a continuous map and End(X) be the number of endpoints of X. We prove the following Theorem 1. Let X be a tree. Then the following holds. (i) Let n > 1 be an integer with no prime divisors less than or equal to End(X) + 1. If a map f : X --> X has a cycle of period n, then f has cycles of all periods greater than 2 End(X)(n - 1). Moreover, h(f) greater-than-or-equal-to ln 2/(n End(X) - 1). (ii) Let 1 less-than-or-equal-to n less-than-or-equal-to End(X) and E be the set of all periods of cycles of some interval map. Then there exists a map f : X --> X such that the set of all periods of cycles of f is {1} or nE, where nE = {nk : k is-an-element-of E}. This implies that if p is the least prime number greater than End(X) and f has cycles of all periods from 1 to 2 End(X)(p - 1), then f has cycles of all periods (for tree maps this verifies Misiurewicz's conjecture, made in Bratislava in 1990). Combining the spectral decomposition theorem for graph maps with our results, we prove the equivalence of the following statements for tree maps: (i) there exists n such that f has a cycle of period mn for any m; (ii) h(f) > 0. Note that Misiurewicz's conjecture and the last result are true for graph maps; an alternative proof of the last result may be also found in a paper by Llibre and Misiurewicz.