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Uzbekistan Tashkent
Section Mathematics
Title Asymptotic distribution of hitting times for critical maps of the circle
Author(-s) Ayupov Sh.A.a, Zhalilov A.A.ab
Affiliations Institute of Mathematics, National Academy of Sciences of Uzbekistana, Yeoju Technical Institute in Tashkentb
Abstract 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
UDC 517.9
MSC 37A05, 28D05
DOI 10.35634/vm210302
Received 24 February 2021
Language English
Citation 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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