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Generalizing the rotation interval to vertex maps on graphs


Authors: Chris Bernhardt and P. Christopher Staecker
Journal: Trans. Amer. Math. Soc. 367 (2015), 4235-4252
MSC (2010): Primary 37E15, 37E25, 37E45
DOI: https://doi.org/10.1090/S0002-9947-2014-06229-6
Published electronically: September 5, 2014
MathSciNet review: 3324926
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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 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.


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Additional Information

Chris Bernhardt
Affiliation: Department of Mathematics and Computer Science, Fairfield University, Fairfield, Connecticut 06824
Email: cbernhardt@fairfield.edu

P. Christopher Staecker
Affiliation: Department of Mathematics and Computer Science, Fairfield University, Fairfield, Connecticut 06824
Email: cstaecker@fairfield.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06229-6
Keywords: Rotation number, rotation interval, graphs, vertex maps, periodic orbits, homotopic to constant map, $\mathbb{Q}$-group
Received by editor(s): September 21, 2012
Received by editor(s) in revised form: February 14, 2013, May 27, 2013, and June 1, 2013
Published electronically: September 5, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.