Multifractal properties of Hao's 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 Hao's work in several directions: (1) We generalize Hao's 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 Hao's sequence representations. (4) So far the connection between the Hao's scheme and another well-known subsequence visualization scheme—Jeffrey's chaos game representation (CGR)—has been characterized only in very vague terms. We show that the fractal dimension results for Hao's visualization frames directly translate to Jeffrey's CGR scheme.