Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences

TL;DR

This paper introduces a wavelet-based analytical framework for studying the ultrametric structure of cyclic symbolic sequences, including a novel formula for counting two-fold de Bruijn sequences, advancing methods in complex systems analysis.

Contribution

It develops a wavelet analysis approach on symbolic sequences and derives a new formula for counting two-fold de Bruijn sequences, extending existing sequence theories.

Findings

Derived a formula for counting two-fold de Bruijn sequences

Demonstrated the wavelet analysis approach on symbolic sequences

Discussed potential advantages in applied contexts

Abstract

The concept of symbolic sequences play important role in study of complex systems. In the work we are interested in ultrametric structure of the set of cyclic sequences naturally arising in theory of dynamical systems. Aimed at construction of analytic and numerical methods for investigation of clusters we introduce operator language on the space of symbolic sequences and propose an approach based on wavelet analysis for study of the cluster hierarchy. The analytic power of the approach is demonstrated by derivation of a formula for counting of {\it two-fold de Bruijn sequences}, the extension of the notion of de Bruijn sequences. Possible advantages of the developed description is also discussed in context of applied p