ON PERIODIC POINTS OF MAPS OF TREES AND THE EXPANSIVE PROPERTY

ABSTRACT. In this paper, it is shown that any map, f, of a tree, T, into itself, where Nn=l,•fn(T) is not a singleton, must have at least two peri-odic points, with period not greater than the number of end points of T. This result is then used to give an elementary proof of a result due to Kato, specifically, that the shift homeomorphism on any non-degenerate inverse limit of a tree with a single bonding map is not expansive. 0. Introduction. Jakobsen and Utz, in 1960 [12], stated, without proof, that the shift homeomorphism on the inverse limit of an arc with any bonding map is not expansive. Karo, in [13], extended this result to trees. The nature of periodic points has a strong relationship to the idea of an expansiv