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

1994; Springer Science+Business Media; Volume: 99; Issue: 2 Linguagem: Inglês

10.1007/bf01199024

1432-2064

Zaqueu Coelho, Pierre Collet,

Topological and Geometric Data Analysis

Given two pointsx, y∈S 1 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 ε>0 the point process obtained by recording the timesn>0 such that |f n (x)−f n (y)|≦ε. With the further assumption that the density of μ is bounded away from zero, we show that when ε tends to zero the above point process scaled by ε−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.