Multifractal properties of Haoʹs geometric representations of DNA sequences

Hao proposed a graphic representation of subsequence structure in DNA sequences and computed fractal dimensions of such representations for factorizable languages. In this study, we extend Haoʹs work in several directions: (1) We generalize Haoʹs scheme to accommodate sequences over an arbitrary finite number of symbols. (2) We establish a direct correspondence between the statistical characterization of symbolic sequences via Rényi entropy spectra and the multifractal characteristics (Rényi generalized dimensions) of the sequences’ spatial representations. (3) We show that for general symbolic dynamical systems, the multifractal fH-spectra in the sequence space endowed with commonly used metrics, coincide with the fH-spectra on Haoʹs sequence representations. (4) So far the connection between the Haoʹs scheme and another well-known subsequence visualization scheme—Jeffreyʹs chaos game representation (CGR)—has been characterized only in very vague terms. We show that the fractal dimension results for Haoʹs visualization frames directly translate to Jeffreyʹs CGR scheme.