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

• Interpretation of the symbolic dynamics of unimodal maps using Gray codes.
• Theoretical framework for estimating the initial conditions/control parameters from the symbolic sequences of unimodal maps.
• Application of the proposed theoretical framework in the context of the cryptanalysis of chaos-based cryptosystems.

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.