Geodesic
Asymptotic distribution of hitting times for critical maps of the circle
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 3, pp. 365-383 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is well known that the renormalization group transformation $\mathcal{R}$ has a unique fixed point $f_{cr}$ in the space of critical $C^{3}$-circle homeomorphisms with one cubic critical point $x_{cr}$ and the golden mean rotation number $\overline{\rho}:=\frac{\sqrt{5}-1}{2}.$ Denote by $Cr(\overline{\rho})$ the set of all critical circle maps $C^{1}$-conjugated to $f_{cr}.$ Let $f\in Cr(\overline{\rho})$ and let $\mu:=\mu_{f}$ be the unique probability invariant measure of $f.$ Fix $\theta \in(0,1).$ For each $n\geq1$ define $c_{n}:=c_{n}(\theta)$ such that $\mu([x_{cr},c_{n}])=\theta\cdot\mu([x_{cr},f^{q_{n}}(x_{cr})]),$ where $q_{n}$ is the first return time of the linear rotation $f_{\overline{\rho}}.$ We study convergence in law of rescaled point process of time hitting. We show that the limit distribution is singular w. r. t. the Lebesgue measure.
Keywords: circle homeomorphism, critical point, rotation number, hitting time, thermodynamic formalism. 
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Sh. A. Ayupov; A. A. Zhalilov. Asymptotic distribution of hitting times for critical maps of the circle. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 3, pp. 365-383. http://geodesic.mathdoc.fr/item/VUU_2021_31_3_a1/

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