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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, Volume 31, Issue 3, Pages 365–383
DOI: https://doi.org/10.35634/vm210302
(Mi vuu775)
 

MATHEMATICS

Asymptotic distribution of hitting times for critical maps of the circle

Sh. A. Ayupova, A. A. Zhalilovba

a Institute of Mathematics of ASRUz, Tashkent, Uzbekistan
b Yeoju Technical Institute in Tashkent, Tashkent, Uzbekistan
References:
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.
Received: 24.02.2021
Bibliographic databases:
Document Type: Article
UDC: 517.9
MSC: 37A05, 28D05
Language: English
Citation: Sh. A. Ayupov, A. A. Zhalilov, “Asymptotic distribution of hitting times for critical maps of the circle”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 31:3 (2021), 365–383
Citation in format AMSBIB
\Bibitem{AyuZha21}
\by Sh.~A.~Ayupov, A.~A.~Zhalilov
\paper Asymptotic distribution of hitting times for critical maps of the circle
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2021
\vol 31
\issue 3
\pages 365--383
\mathnet{http://mi.mathnet.ru/vuu775}
\crossref{https://doi.org/10.35634/vm210302}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000702444500002}
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    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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