On Hitting Times of Circle Maps with Generalized Dynamical Partitions

Abstract

In present work we study the hitting times for circle homeomorphisms with one break point and universal renormalization properties. Consider the set $X(\rho) $ of all orientation preserving circle homeomorphisms $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_{b}\}),\,\,\,\varepsilon>0,$ with one break point $ x_{b}$ and irrational rotation number $\rho_{T}=\frac{-k+\sqrt{{{k}^{2}}+4}}{2},\,\,k\ge 1.$
For each $n\geq1$ we define $c_{n}:=c_{n}(c)$ such that $\mu([x_{b},c_{n}])=c\cdot\mu([x_{b},T^{q_{n}}(x_{b})]),$ where $q_{n}$
are first return times of $T.$ Denote by $E_{n,c}(x)$ first hitting times of $x$ to interval $[x_{b},c_{n}]. $ Consider the rescaled first hitting time $\overline{E}_{n,c}:=\frac{1}{q_{n+1}}E_{n,c}(x)$. We study convergence in law of random variables $\overline{E}_{n,c}(x).$ We show that the limit distribution is singular w.r.t. Lebesgue measure.