Abstract:
In this paper, we investigate the relationship between local maps and global maps.
Let  be a cellular automaton with  a cell space, S a state set, N a neighborhood of 0 in  Further, f is an arbitrarily chosen element in  called a local map,  the configuration set and F in  is a global map.
Then we find a correspondence between the set of local maps and a certain subset  of global maps in .
First in Theorem A, we construct a bijective correspondence between  and 
Next, in Theorem B, assuming that S is a topological space, we show that there exists a homeomorphism between two topological spaces  and
In Theorem C, we show a bijective correspondence between the set of continuous local maps f and the set of continuous global maps F.
In Theorem D, in case that S is a discrete topological space, we show that the following two conditions for F are equivalent:
(a) F is continuous and shift commutative.
(b) F is a global map of some local map f with respect to some subset N of 
This is a generalization of Richardson’s Theorem where S and N were assumed to be finite.
Finally, in Theorem E, under the assumption that S is a discrete topological space, we provide two different necessary and sufficient conditions for  to contain only continuous local maps.
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| Keywords and phrases: cellular automaton, local map, global map, configuration, Richardson’ theorem.
Received: March 14, 2023; Accepted: May 2, 2023; Published: July 19, 2023
How to cite this article: Hiroyuki Ishibashi, Correspondence between local maps and global maps in cellular automata, JP Journal of Geometry and Topology 29(2) (2023), 121-151. http://dx.doi.org/10.17654/0972415X23006
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License  
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