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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
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
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
Linking options:
https://www.mathnet.ru/eng/vuu775 https://www.mathnet.ru/eng/vuu/v31/i3/p365
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Abstract page: | 217 | Full-text PDF : | 91 | References: | 30 |
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