Ultrametric arrangement of symbolic sequences
Symbolic sequence is a fundamental object, which plays important role in mathematical modelling in various areas of modern science and technology. In the talk we introduce a notion of $p$-close sequences and utilize it to construct an ultrametric distance on the set of symbolic sequences of a given length. We show, that the sequences are organized into hierarchically nested clusters with irregular brunching parameter. We study statistics of clusters using equivalence of the problem to the one of counting degenerates in the length spectrum of de Bruijn graphs (a standard mathematical tool for visualization of sequences) and employ methods of random matrix theory for calculation of the obtained multidimensional integrals. In the talk a special attention will be payed to such applied problems as (i) classification of periodic orbits in quantum chaotic systems for calculation of spectral correlations and (ii) choice of parameters in the DNA sequence assembling algorithms.