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. 1. If n is not a divisor of 2 ^k then f has a periodic point with period 2 ^k .

   2. If , where is odd and , then f has a periodic point with period 2 ^p r for any .

   3. 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.

Keywords:

• tree maps
• periods of orbits
• Sharkovsky's theorem
• binary expansion

1991 Mathematics Subject Classification::

• 37E05