Shift spaces, Languages and Transfinite Induction

TL;DR

This paper extends the classical concept of shift spaces to uncountable index sets, introducing a generalized language and characterization using transfinite induction, broadening the theoretical framework of symbolic dynamics.

Contribution

It introduces a generalized framework for shift spaces over uncountable index sets, incorporating a new notion of language and a characterization method using transfinite induction.

Findings

Generalization of shift spaces to uncountable index sets

Introduction of a new language concept for extended shift spaces

Use of transfinite induction for characterization

Abstract

This paper deals with an extension of the classical concept of shift space, which corresponds to any shift-invariant closed subset of the Cartesian product of a particular finite set (alphabet) endowed with the prodiscrete topology. In such an extended framework the notion of language is introduced and a characterization is shown. In order to do this, transfinite induction is required because the cardinality of the index set of the product may not be countable.