Periods of orbits for maps on graphs homotopic to a constant map

TL;DR

This paper characterizes the set of periods for maps on graphs homotopic to a constant map with a periodic orbit at vertices, revealing specific conditions for the existence of periodic points with certain periods.

Contribution

It establishes new theorems describing the periods of orbits for such graph maps, extending understanding of periodic behavior in this setting.

Findings

If v is not a divisor of 2^k, then a period 2^k point exists.

For v=2^ks with odd s>1, periodic points of period 2^k r exist for all r>s.

Results relate to Sharkovsky ordering of integers.

Abstract

The paper proves two theorems concerning the set of periods of periodic orbits for maps of graphs that are homotopic to the constant map and such that the vertices form a periodic orbit. The first result is that if $v$ is not a divisor of $2^k$ then there must be a periodic point with period $2^k$. The second is that if $v=2^ks$ for odd $s>1$, then for all $r>s$ there exists a periodic point of minimum period $2^k r$. These results are then compared to the Sharkovsky ordering of the positive integers. (The final version of this paper will appear in the Journal of Difference Equations and Applications.)