2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79816The mechanism of the exponential transient statistics of Poincaré 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 Poincaré recurrences including the transition from exponential to exponential statistics.Latex 2.09, 8 figuresChaotic Dynamicstext