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

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 twofold 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 problem of construction of efficient DNA sequence assembly algorithms. 1. Motivation and structure. Nowadays symbolic sequences is a fundamental concept widely used in various fields of natural sciences [1–9]. In bioinformatics, theory of information, and theory of discreet Markov chains the intrinsic structure of objects under consideration provides direct mapping on symbolic sequences [1]. Less obvious extension of the symbolic approach one can find in study of complex behavior in dynamical systems [2,3]. In essence, the underlying idea is in organization of a stroboscopic sampling of the multidimensional trajectory. In case of Hamiltonian systems one constructs a Poincaré section surface in phase-space [10], such that it is oriented orthogonal to the dynamical flow at each point. Linearization of the dynamics allows to separate stable and unstable directions of motion and thus define the set of feasible positions at the next crossing of the Poincaré section surface. All intersections falling within the same sub-region of the surface are designated by a certain symbol. For the system possessing a chaotic dynamics one can aver existence of the Markov partitioning of the surface, meaning that each next symbol in the symbolic dynamics is defined only by the previous one. For infinite or cyclic sequences the real trajectory can be uniquely restored from the symbolic dynamics. Often, the obtained Markov alphabet includes an infinite number of symbols, as, for instance, in chaotic billiards, out of the flow discontinuity on the billiard walls. It is