Section
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Mathematics
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Title
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Asymptotic distribution of hitting times for critical maps of the circle
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Author(-s)
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Ayupov Sh.A.a,
Zhalilov A.A.ab
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Affiliations
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Institute of Mathematics, National Academy of Sciences of Uzbekistana,
Yeoju Technical Institute in Tashkentb
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Abstract
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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.
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Keywords
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circle homeomorphism, critical point, rotation number, hitting time, thermodynamic formalism
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UDC
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517.9
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MSC
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37A05, 28D05
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DOI
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10.35634/vm210302
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Received
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24 February 2021
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Language
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English
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Citation
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Ayupov Sh.A., Zhalilov A.A. Asymptotic distribution of hitting times for critical maps of the circle, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 3, pp. 365-383.
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