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Summary Given two pointsx, y?S1 randomly chosen independently by a mixing absolutely continuous invariant measure µ of a piecewise expanding and smooth mapf of the circle, we consider for each e>0 the point process obtained by recording the timesn>0 such that |fn(x)-fn(y)|?e. With the further assumption that the density of µ is bounded away from zero, we show that when e tends to zero the above point process scaled by e-1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle