Usatenko O. Correlation properties of random discrete dynamical systems

A mathematical tool for dichotomous multi-step additive Markov chains is developed and their main statistical characteristics are investigated. The equation connecting the correlation and memory functions is derived. Methods for constructing multi-step dichotomous chains with a given correlation function are proposed. Propagation of electromagnetic waves in a chain of Josephson junctions with correlated disorder is studied. It is shown that the localization length of electrons in the exon parts of the DNA is much higher than their intron localization length. A method for constructing random diffraction gratings with long-range correlations, which have given diffraction properties is developed. Nonextensive thermodynamic properties of spin chain with arbitrary, but finite interaction range are studied. Using the developed theory of the multi-step Markov chains leads us at first glance to a number of paradoxical conclusions about the statistical properties of natural language texts and DNA sequences. The self-similarity property of literary texts is found. The same property manifests itself in the dichotomous model of additive multi-step chain with a step-wise memory function.