Poincar\'e recurrences of DNA sequence

We analyze the statistical properties of Poincaré recurrences of Homo sapiens, mammalian and other DNA sequences taken from Ensembl Genome data base with up to fifteen billions base pairs. We show that the probability of Poincaré recurrences decays in an algebraic way with the Poincaré exponent β ≈ 4 even if oscillatory dependence is well pronounced. The correlations between recurrences decay with an exponent ν ≈ 0.6 that leads to an anomalous super-diffusive walk. However, for Homo sapiens sequences, with the largest available statistics, the diffusion coefficient converges to a finite value on distances larger than million base pairs. We argue that the approach based on Poncaré recurrences determines new proximity features between different species and shed a new light on their evolution history. The Poincaré recurrence theorem of 1890 [1] states that after a certain time a dynamical Hamiltonian trajectory in a bounded phase space always returns to a close vicinity of an initial state. Even if recurrences definitely take place the question about their properties, or more exactly what are the statistics of Poincaré recurrences, and what are their correlation properties, still remain an unsolved problem for systems of dynamical chaos even after an impressive development of the theory of dynamical complexity [2–4]. The two limiting case of periodic and fully chaotic motion are well understood: in the first case the recurrences are periodic while in the latter case the probability of recurrences P (t) with time being larger than t drops exponentially at t → ∞ [2–4]. Thus, the latter case is similar to a coin flipping, where a probability to stay on the same side after more than t flips decays at 2 −t. However, in generic Hamiltonian systems the probability P (t) decays algebraically with t, as P (t) ∼ 1/t β , due to long trappings in a vicinity of stability islands showing the Poincaré exponent β ≈ 1.5 [5–10]. A detailed theoretical explication of this slow algebraic decay is still lacking. Usually, the consecutive recurrences in dynam-ical systems are not correlated since a trajectory passes across domains of chaotic component. The Poincaré recurrences represent a powerful tool for analysis of statistical properties of symbolic trajectories of various types [2–4]. Surprisingly, this powerful tool of dynamical systems has not been applied for detailed statistical studies of DNA sequence which also can be viewed as a symbolic trajectory. There have been only a few earlier …