Poincar\'e Recurrences in Microtron and the Global Critical Structure

The mechanism of the exponential transient statistics of Poincar\'e recurrences in the presence of chaos border with its critical structure is studied using two simple models: separatrix map and the kicked rotator ('microtron'). For the exponential transient to exist the two conditions have been shown to be crucial: fast (ballistic) relaxation, and a small measure of the critical structure. The latter was found to include a new peripheral part (halo) of a surprisingly large size. First preliminary empirical evidence is presented for a new regime of Poincar\'e recurrences including the transition from exponential to exponential statistics.