Fixed point sets in digital topology, 1


  • Laurence Boxer Niagara University
  • P. Christopher Staecker Fairfield University



digital image, fixed point, retraction


In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.

We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(Cn) where Cn is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including Cn, in which F(X) does not equal {0, 1, . . . , #X}.

We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.


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Author Biographies

Laurence Boxer, Niagara University

Department of Computer and Information Sciences

P. Christopher Staecker, Fairfield University

Department of Mathematics


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How to Cite

L. Boxer and P. C. Staecker, “Fixed point sets in digital topology, 1”, Appl. Gen. Topol., vol. 21, no. 1, pp. 87–110, Apr. 2020.



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