Abstract
In the last sections we realized that the Cantor metric and cellular automata fit very well together: any cellular automaton can be viewed as a continuous map of a metric Cantor space and any continuous function on a metric Cantor space can be embedded into a cellular automaton. However, there is one major drawback: the Cantor metric is not translational invariant. The metric is focused on a region near the origin and everything far away is neglected. The amount of information that is neglected may be tremendously large, as “everything else” is an infinite region, while the “near region” is finite. A perturbation of a state is considered small if it agrees with the state on a (large, but finite) region around the origin. The states at cells in the remaining, infinite part of the grid may be arbitrary. The intuitive idea of a “small perturbation” is not met by the concept of the Cantor metric.
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Hadeler, KP., Müller, J. (2017). Besicovitch and Weyl Topologies. In: Cellular Automata: Analysis and Applications. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-53043-7_4
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DOI: https://doi.org/10.1007/978-3-319-53043-7_4
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