Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps

TL;DR

This paper formalizes the use of Gray codes in symbolic dynamics of unimodal maps, clarifying theoretical connections, correcting inaccuracies, and providing a rigorous mathematical framework for previous heuristic approaches.

Contribution

It offers a formal mathematical formulation linking Gray codes with symbolic dynamics of unimodal maps, improving theoretical understanding and correcting prior inaccuracies.

Findings

Established a rigorous mathematical connection between Gray codes and symbolic dynamics.

Reinterpreted classical symbolic dynamics theory for unimodal maps using Gray codes.

Identified and corrected inaccuracies in previous symbolic dynamics literature.

Abstract

In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.