The Loss of Tightness of Time Distributions for Homeomorphisms of the Circle

For a minimal circle homeomorphism f we study convergence in law of rescaled hitting time point process of an interval of length ε > 0. Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence εn going to zero exhibiting coexistence of two non-trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.