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Shift spaces, Languages and Transfinite Induction

Maira Aranguren, Jorge Campos, Neptalí Romero, Ramón Vivas

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.

Shift spaces, Languages and Transfinite Induction

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.
Paper Structure (4 sections, 7 theorems, 11 equations)

This paper contains 4 sections, 7 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. $\mathbb{H}$-shift spaces and their languages
  3. Conclusions
  4. Acknowledgements:

Key Result

Theorem 1.1

If $\mathcal{L}$ is a subset of $\mathcal{A}^*$ with properties $L_1$ and $L_2$, then there exists a shift space $X\subset\mathcal{A}^\mathbb{Z}$ such that $\mathcal{L}(X)=\mathcal{L}$.

Theorems & Definitions (20)

  • Theorem 1.1: lind
  • Definition 2.1
  • Example 2.1
  • Example 2.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 10 more