Limiting Laws for Entrance Times of Critical Mappings of a Circle

Abstract

A renormalization group transformation R1 has a single stable point \(T_{\xi _0 ,\eta _0 } \) in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number \(\rho = (\sqrt 5 - 1)/2\) (“the golden mean”). Let a homeomorphism T be the C1-conjugate of \(T_{\xi _0 ,\eta _0 } \). We let \(\{ \Phi _n^{(k)} (t),n = \overline {1,\infty } \} \) denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any \(t \in \mathbb{R}^1 \), the sequence \(\{ \Phi _n^{(1)} (t)\} \) has a finite limiting distribution function \(\Phi ^{(1)} (t)\), which is continuous in \(\mathbb{R}^1 \), and singular on the interval [0,1]. We also study the sequence \(\{ \Phi _n^{(k)} (t),n = \overline {1,\infty } \} \) for k > 1.