Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation $\mathbf R_1$ has a single stable point 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 $C^{1}$-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 $k$th entrance to the $n$th 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 $\Phi_n^{(1)}(t)$, which is continuous in $\mathbb{R}^1$, and singular on the interval $[0,1]$. We also study the sequence $\bigl\{\Phi_{n}^{(k)}(t), \ n=\overline{1,\infty}\bigr\}$ for $k>1$.