Abstract:
We consider the circle homeomorphism $f\in {{C}^{2+\varepsilon }}({{S}^{1}}\backslash \{b\})$, $\varepsilon > 0$, with one break point $b$ and irrational rotation number $\rho ={{\rho }_{f}}=\frac{\sqrt{5}-1}{2}$. Let ${{q}_{n}}$ be the first return time. We fix arbitrary point ${{z}_{0}}\in {{S}^{1}}$. We denote by ${{J}_{n}}({{z}_{0}})$ the $n$-th renormalization interval of ${{z}_{0}}$. Let $\bar{E}_{n}^{(1)}(x)$ be the normalized entrance time function. The distribution function of random variable $\bar{E}_{n}^{(1)}(x)$ to Lebesgue measure $l$ denote by $\Phi _{n}^{(1)}(t)$. We prove that $\Phi _{n}^{(1)}(t)\to \Phi (t)$, $n\to \infty$ for all $t\in {{\mathbb{R}}^{1}}$ and $\Phi_{n}^{(1)}(t)$ is singular on $[0,1]$ w.r.t. Lebesgue measure $l$.
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