We give a condition for nonperiodic rank-one systems to have non-exponential asymptotic distribution (equal to 1[1,∞[) of return times along subsequences of cylinders. Applying this result to the staircase transformation, we derive mixing dynamical systems with non-exponential asymptotics. Moreover, we show for two columns rank-one systems unique asymptotic along full sequences of cylinders.

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We answer to questions about n-tuple of return time limit laws raised in [2]. By constructing rank-one systems with a 2-tuple different from (F, F ) along sequence of cylinders, we prove that the second return time limit law is not entirely determined by the first return time limit law. Matrix technique is developed in order to solve the combinatorial problem of 2-tuple return time laws. Thus, using a good implementation of a sequence of periodic systems in a rank-one system, examples of rank-one systems such that the 2-tuple return time limit laws along sequences of cylinders satisfiesF (1) =F (2) are given.