A Curtis-Hedlund-Lyndon theorem for Besicovitch and Weyl spaces

Authors:
Johannes Müller;Christoph Spandl
Affiliations:
Technical University Munich, Centre for Mathematical Sciences, Boltzmannstr. 3, D-85748 Garching/Munich, Germany;Institute for Theoretical Computer Science und Mathematics, University of the Federal Armed Forces Munich, D-85577 Neubiberg, Germany
Venue:
Theoretical Computer Science
Year:
2009

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Abstract

Global functions of cellular automata on state spaces equipped with the Cantor topology are well characterized by the Curtis-Hedlund-Lyndon theorem. In this paper, we develop a characterization of global functions of cellular automata on Z, if the state space is equipped by Weyl and Besicovitch topology. The necessary and sufficient condition for a function to be the global map of a cellular automaton is (1) a strong localization property, a condition that strengthen Lipschitz continuity, (2) the set of (Cantor) periodic states are positively invariant and (3) the function commutes (in the Weyl/Besicovitch sense) with the shift operator.