Weak convergence of hitting times for critical circle maps

Abstract

Let Cr(ρ) be the set of all critical circle which maps are C 1 conjugate to f cr ∈ C 3 critical circle homeomorphisms having a single x cr critical point and rotation number ρ := [k,k,k,,...]. Let µ := µ f denote the unique probability invariant measure of the map f ∈ Cr(ρ). Define a decreasing sequence {c n := c n (θ), n ≥ 1} for some θ ∈ (0,1) is such that a µ−measure of the interval (x cr ,c n ] satisfies µ([x cr ,c n ]) = θ ·µ([x cr ,f q n (x cr )]), where q n is the return times associated with the linear rotation f ρ = x+ρmod1. We study weak convergence of normalized hitting times. Moreover, we show that limiting distribution is singular with respect to the Lebesgue measure