A Sharkovsky theorem for vertex maps on trees

Abstract

Let T be a tree with n vertices. Let be continuous and suppose that the n vertices form a periodic orbit under f. We show: 1. a. If n is not a divisor of 2 k then f has a periodic point with period 2 k . b. If , where is odd and , then f has a periodic point with period 2 p r for any . c. The map f also has periodic orbits of any period m where m can be obtained from n by removing ones from the right of the binary expansion of n and changing them to zeroes. 2. Conversely, given any n, there is a tree with n vertices and a map f such that the vertices form a periodic orbit and f has no other periods apart from the ones given above.