Abstract
We consider a model of a billiard-type system, which consists of two chambers connected through a hole. One chamber has a circle-shaped scatterer inside (Sinai billiard with infinite horizon), and the other one has a Cassini oval with a concave border. The phase space of the Cassini billiard contains islands, and its parameters are taken in such a way as to produce a self-similar island hierarchy. Poincar\'e recurrences to the left and to the right chambers are considered. It is shown that the corresponding distribution function does not reach ``equipartition'' even during the time ${10}^{10}$. The explanation is based on the existence of singularities in the phase space, which induces anomalous kinetics. The analogy to the Maxwell's Demon model is discussed.
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