The Poincar\'e recurrence time for the de Sitter space with dynamical chaos

Abstract

For an ordinary thermodynamical system the Poincar\'{e} recurrence time is exponentially large in the Boltzmann entropy of the system. It turns out, that for a system with dynamical chaos it is determined by the Kolmogorov-Sinai entropy and can be considerably shorter. It is shown in this note that for the de Sitter space with strong mixing properties the mean recurrence time is equal to the inverse Hubble constant. This means that our universe can have a finite lifetime bounded by the current age of the universe. After this time, the universe should recycle itself and this process has neither a beginning nor an end.