Poincaré recurrences and correlations decay in the systems with divided phase space

The investigations of the Poincaré recurrences in the systems with the divided phase space (for example, Chirikov standard map) showed that the probability $P$ to stay in some part of phase space during time larger than $\tau$ decay in a power law $P(\tau) \sim 1/{\tau^{p}}$ with the exponent $p < 2$. The average value of $p$ is 1.5 but in some cases its value can be 1.3 or even very close (but larger) to 1 (refs.[18,30]). Such slow power decay is quite different from exponential decay typical for completely chaotic systems (for example, Anosov systems). The decay of correlations $C$ is connected with the decay of $P$ in the way: $C(\tau) \sim P(\tau) \tau \sim 1/{\tau^{p-1}}$. Since $p < 2$ this mean that such slow correlations decay can lead to a divergence of integral of correlations giving in some cases infinite diffusion rate (streaming). Such divergence indeed exists in the standard map for parameter values when there exist accelerating modes (however, the dynamics is considered for trajectories in the chaotic component). For the first time such anomalously slow decay of $P(\tau)$ was observed in [5]. Later it was investigated also by Karney and many others. It was shown that this slow decay is originated by the hierarchical renormgroup structure of the phase space [5,18,30]. The investigations of refs. [18,30] showed that the value of $p$ becomes closer to 1 if the chaotic component is restricted by critical golden invariant curve. However, in such case for different maps different values of $p \approx 1; 1.3$ appear that looks to be in some contradiction with the universal structure of critical golden curve. The theory developed by Ott and Meiss was not able to explain the numerical values of $p$ close to 1. Futher investigations are required for a better understanding of Poincaré recurrences. However, recent results obtained in [101] showed that for asymptotically large times the exponent $p$ approaches to the value $p=3$ (see more details in the section 3.6.1).