The Besicovitch pseudometric is a shift-invariant pseudometric over the set of infinite sequences, that enjoys interesting properties and is suitable for studying the dynamics of cellular automata. It corresponds to the asymptotic behavior of the Hamming distance on longer and longer prefixes. Though dynamics of cellular automata were already studied in the literature, we propose the first study of the dynamics of substitutions. We characterize those that yield a well-defined dynamical system as essentially the uniform ones. We also explore a variant of this pseudometric, the Feldman–Katok pseudometric, where the Hamming distance is replaced by the Levenshtein distance. Like in the Besicovitch space, cellular automata are Lipschitz in this space, but here also all substitutions are Lipschitz. In both spaces, we discuss equicontinuity of these systems, give a number of examples, and generalize our results to the class of dill maps, that embeds both cellular automata and substitutions.

贝西科维奇伪距是无限序列集合上的平移不变伪距，具有有趣的性质，适合研究元胞自动机的动力学。它对应于汉明距离在越来越长的前缀上的渐近行为。尽管元胞自动机的动力学已经在文献中进行了研究，但我们提出了对替代动力学的第一个研究。我们将那些产生明确定义的动力系统的特征描述为本质上一致的系统。我们还探索了这种伪度量的一个变体，即 Feldman-Katok 伪度量，其中汉明距离被 Levenshtein 距离取代。与贝西科维奇空间一样，元胞自动机在该空间中是利普希茨（Lipschitz），但这里所有的替换也是利普希茨（Lipschitz）。在这两个空间中，