ASYMPTOTIC DISTRIBUTION OF HITTING TIMES FOR CRITICAL MAPS OF THE CIRCLE

It is well known that the renormalization group transformation R has a unique fixed point f(cr) in the space of critical C-3-circle homeomorphisms with one cubic critical point x(cr) and the golden mean rotation number (rho) over bar := root 5-12. Denote by Cr((rho) over bar) the set of all critical circle maps C-1-conjugated to f(cr). Let f is an element of Cr((rho) over bar) and let mu := mu(f) be the unique probability invariant measure of f. Fix theta is an element of(0, 1). For each n >= 1 define c(n) := c(n) (theta) such that mu([x(cr), c(n)]) = theta center dot mu([x(cr), f(qn) (x theta(cr))]), where q(n) is the first return time of the linear rotation f((rho) over bar). We study convergence in law of rescaled point process of time hitting. We show that the limit distribution is singular w. r. t. the Lebesgue measure.