Generalizing the rotation interval to vertex maps on graphs

TL;DR

This paper extends the concept of rotation intervals from circle maps to vertex maps on graphs, analyzing periodic points and their rotation elements to reveal infinite families of related periodic points.

Contribution

It introduces a generalization of rotation intervals to graph maps homotopic to the identity, linking periodic points and their rotation elements.

Findings

Existence of two periodic points with certain rotation elements implies infinite related periodic points.

Results generalize rotation interval concepts from circle maps to graph maps.

Provides new insights into periodic point structure in graph dynamics.

Abstract

Graph maps that are homotopic to the identity and that permute the vertices are studied. Given a periodic point for such a map, a {\em rotation element} is defined in terms of the fundamental group. A number of results are proved about the rotation elements associated to periodic points in a given edge of the graph. Most of the results show that the existence of two periodic points with certain rotation elements will imply an infinite family of other periodic points with related rotation elements. These results for periodic points can be considered as generalizations of the rotation interval for degree one maps of the circle.