ASYMPTOTIC LIMIT LAW FOR THE CLOSE APPROACH OF 2 TRAJECTORIES IN EXPANDING MAPS OF THE CIRCLE

Given two points x, y is-an-element-of S1 randomly chosen independently by a mixing absolutely continuous invariant measure u of a piecewise expanding and smooth map f of the circle, we consider for each epsilon > 0 the point process obtained by recording the times n > 0 such that \f(n)(x)-f(n)(y)\ less-than-or-equal-to epsilon. With the further assumption that the density of u is bounded away from zero, we show that when epsilon tends to zero the above point process scaled by epsilon-1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicitly by an average on the rate of expansion of f.