ISRAEL JOURNAL OF MATHEMATICS 9S (1996), 93-112 LIMIT LAWS OF ENTRANCE TIMES FOR HOMEOMORPHISMS OF THE CIRCLE BY ZAQUEU COELHO AND EDSON DE FARIA* Instituto de Matemdtica e Estatlstica, Universidade de S(~o Paulo Caixa Postal 20570, 01,~52-990 Sdo Paulo SP, Brazil Z. Coelho e-mail: zaqueu@ime.usp.br, E. de Faria e-mail: edson@ime.usp.br ABSTRACT Given a homeomorphism f of the circle with irrational rotation number and a descending chain of renormalization intervals Yn of f, we consider for each interval the point process obtained by marking the times for the orbit of a point in the circle to enter Yn. Assuming the point is randomly cho- sen by the unique invariant probability measure of f, we obtain necessary and sufficientconditions which guarantee convergence in law of the corre- sponding point process and we describe all the limiting processes. These conditions are given in terms of the convergent subsequences of the orbit of the rotation number of f under the Gauss transformation and under a certain realization of its natural extension. We also consider the case when the point is randomly chosen according to Lebesgue measure, f be- ing a diffeomorphism which is Cl-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case. Introduction Limit laws of entrance times have been obtained in various contexts such as: hyperbolic automorphisms of the torus and Markov chains [Pi], Axiom A diffeo- morphisms and shifts of finite type with a HSlder potential [Hi], and piecewise expanding maps of the circle [CG] (see also [CC]). The general setting for the problem is as follows. Given an ergodic dynamical system (X, B, #, f) and a set * This work is part of Projeto Tem~tico de Equipe "Transi~o de Fase Din~mica em Sistemas Evolutivos" financially supported by FAPESP grant 90/3918-5. Received March 16, 1994 93