Vol. 10, No. 3, 2017

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Discrete dynamics of contractions on graphs

Olena Ostapyuk and Mark Ronnenberg

Vol. 10 (2017), No. 3, 495–503
Abstract

We study the dynamical behavior of functions on vertices of a graph that are contractions in the graph metric. We show that the fixed point set of such functions must be convex. If a function has no fixed points and the graph is a tree, we prove that every dynamical cycle must have an even period and the function behaves eventually like a symmetry.

Keywords
discrete dynamics, dynamics of contractions, graphs
Mathematical Subject Classification 2010
Primary: 39B12, 54H20
Secondary: 05C05
Milestones
Received: 14 February 2016
Revised: 12 April 2016
Accepted: 14 April 2016
Published: 14 December 2016

Communicated by Martin Bohner
Authors
Olena Ostapyuk
Department of Mathematics
University of Northern Iowa
Cedar Falls, IA 50614-0506
United States
Mark Ronnenberg
Department of Mathematics
University of Northern Iowa
Cedar Falls, IA 50614-0506
United States