Cold and freezing sets in the digital plane
Keywords:
digital topology, digital image, approximate fixed point, freezing set, cold setAbstract
Cold sets and freezing sets belong to the theory of (approximate) fixed points for continuous self-maps on digital images. We study some properties of cold sets for digital images in the digital plane, and we examine some relationships between cold sets and freezing sets.
References
L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839.
L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62.
L. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18 (2) (2017), 401-427.
L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19 (1) (2018), 21-53.
L. Boxer, Approximate fixed point properties in digital topology, Bulletin of the International Mathematical Virtual Institute 10 (2) (2020), 357-367.
L. Boxer, Approximate fixed point property for digital trees and products, Bulletin of the International Mathematical Virtual Institute 10(3) (2020), 595-602.
L. Boxer, Fixed point sets in digital topology, 2, Applied General Topology 21(1) (2020), 111-133.
L. Boxer, Convexity and freezing sets in digital topology, Applied General Topology 22 (1) (2021), 121-137.
L. Boxer, Subsets and freezing sets in the digital plane, Hacettepe Journal of Mathematics and Statistics 50 (4) (2021), 991-1001.
L. Boxer, Some consequences of restrictions on digitally continuous functions, submitted. Available at https://arxiv.org/submit/4163172
L. Boxer, O. Ege, I. Karaca, J. Lopez, and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17(2) (2016), 159-172.
L. Boxer and P.C. Staecker, Fixed point sets in digital topology, 1, Applied General Topology 21 (1) (2020), 87-110.
L. Chen, Gradually varied surface and its optimal uniform approximation, SPIE Proceedings 2182 (1994), 300-307.
L. Chen, Discrete Surfaces and Manifolds, Scientific Practical Computing, Rockville, MD, 2004.
J. Haarmann, M.P. Murphy, C.S. Peters, and P.C. Staecker, Homotopy equivalence in finite digital images, Journal of Mathematical Imaging and Vision 53 (2015), 288-302.
S-E Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91.
J.M. Kang and S-E. Han, The product property of the almost fixed point property for digital spaces, AIMS Mathematics 6 (7) (2021), 7215-7228.
T.Y. Kong and R. Kopperman, Digital topology, in K.P. Hart, Jun-iti Nagata, J.E. Vaughan, eds., Encyclopedia of General Topology, Elsevier, 2004.
T.Y. Kong, R. Kopperman, and P.R. Meyer, A topological approach to digital topology, The American Mathematical Monthly 98 (10) (1991), 901-917.
A. Rosenfeld, Digital topology, The American Mathematical Monthly 86 (8) (1979), 621-630.
A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4, 177-184, 1986.
R. Tsaur and M.B. Smyth, 'Continuous' multifunctions in discrete spaces with applications to fixed point theory, In: G. Bertrand, A. Imiya, R. Klette (eds.), Digital and Image Geometry, Lecture Notes in Computer Science, vol. 2243, 151162. Springer, Berlin (2001), doi:10.1007/3-540-45576-05
