Poincar\'e recurrences of DNA sequence

We analyze the statistical properties of Poincar\'e 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 Poincar\'e recurrences decays in an algebraic way with the Poincar\'e exponent $\beta \approx 4$ even if oscillatory dependence is well pronounced. The correlations between recurrences decay with an exponent $\nu \approx 0.6$ 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 Poncar\'e recurrences determines new proximity features between different species and shed a new light on their evolution history.