Sliding block codes between shift spaces over infinite alphabets

A note on the definition of sliding block codes and the Curtis-Hedlund-Lyndon Theorem

In this note we propose an alternative definition for sliding block codes between shift spaces. This definition coincides with the usual definition in the case that the shift space is defined on a finite alphabet, but it encompass a larger class of maps when the alphabet is infinite. In any case, the proposed definition keeps the idea that a sliding block code is a map with a local rule. Using this new definition we prove that the Curtis-Hedlund-Lyndon Theorem always holds for shift spaces over countable alphabets.

Inverse semigroup shifts over countable alphabets

In this work we characterize shift spaces over infinite countable alphabets that can be endowed with an inverse semigroup operation. We give sufficient conditions under which zero-dimensional inverse semigroups can be recoded as shift spaces whose correspondent inverse semigroup operation is a 1-block operation, that is, it arises from a group operation on the alphabet. Motivated by this, we go on to study block operations on shift spaces and, in the end, we prove our main theorem, which states that Markovian shift spaces, which can be endowed with a 1-block inverse semigroup operation, are co…

Local homeomorphisms that *-commute with the shift

Exel and Renault proved that a sliding block code on a one-sided shift space coming from a progressive block map is a local homeomorphism. We provide a counterexample showing that the converse does not hold. We use this example to generalize the notion of progressive to a property of block maps we call weakly progressive, and we prove that a sliding block code coming from a weakly progressive block map is a local homeomorphism. We also introduce the notion of a regressive block map and prove that a sliding block code *-commutes with the shift map if and only if it comes from a regressive block…

One-sided shift spaces over infinite alphabets

We define a notion of (one-sided) shift spaces over infinite alphabets. Unlike many previous approaches to shift spaces over countable alphabets, our shift spaces are compact Hausdorff spaces. We examine shift morphisms between these shift spaces, and identify three distinct classes that generalize the shifts of finite type. We show that when our shift spaces satisfy a property that we call "row-finite", then shift morphisms on them may be identified with sliding block codes. As applications, we show that if two (possibly infinite) directed graphs have edge shifts that are conjugate, then the …

通过置换滑动块码提升严格遍历子转移

Lifts of strictly ergodic subshifts by permutative sliding block codes

该研究给出置换滑动块码提升子转移时保持极小性与唯一遍历性的判据，通过示例构造严格遍历有限到一扩张，还发现其提升的极小子转移可能非唯一遍历的特殊现象。

通过滑动块码编码子移位

Encoding subshifts through sliding block codes

本文在零误差信息论视角下推广Krieger嵌入定理，给出低熵子移位可嵌入有限型混合移位并保持与sofic移位满射码复合后单射的充要条件。