Limit laws of entrance times for homeomorphisms of the circle

Given a homeomorphism f of the circle with irrational rotation number and a descending chain of renormalization intervals J(n) 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 J(n). Assuming the point is randomly chosen by the unique invariant probability measure of f, we obtain necessary and sufficient conditions which guarantee convergence in law of the corresponding 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 being a diffeomorphism which is C-1-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case.