Asymptotic Limit Law for the Close Approach of Two Trajectories in Expanding Maps of the Circle

. Given two points x; y 2 S 1 randomly chosen independently by a mixing absolutely continuous invariant measure ¯ of a piecewise expanding and smooth map f of the circle, we consider for each " ? 0 the point process obtained by recording the times n ? 0 such that jf n (x) \Gamma f n (y)j ". With the further assumption that the density of ¯ is bounded away from zero, we show that when " tends to zero the above point process scaled by " \Gamma1 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 . Mathematics Subject Classification: 60F05, 60G55, 34C35. Z.Coelho e-mail: zaqueu@ime.usp.br P.Collet e-mail: collet@orphee.polytechnique.fr 1 Instituto de Matem'atica e Estat'istica, Universidade de S~ao Paulo, Caixa Postal 20570 (Ag. Iguatemi), 01498 S~ao Paulo SP (Brazil). 2 Centre de Physique Th'eorique, Ecole Polytechnique, F-91128 Palaiseau Cedex (France) Labo..