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Limiting Laws for Entrance Times of Critical Mappings of a Circle

Authors:	Dzhalilov A A

Institution:	(1) Samarkand State University, Samarkand, Uzbekistan

Abstract:	A renormalization group transformation R ₁ has a single stable point $T_{\xi _0 ,\eta _0 }$ in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number $\rho = (\sqrt 5 - 1)/2$ (the golden mean). Let a homeomorphism T be the C ¹-conjugate of $T_{\xi _0 ,\eta _0 }$ . We let $\{ \Phi _n^{(k)} (t),n = \overline {1,\infty } \}$ denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any $t \in \mathbb{R}^1$ , the sequence $\{ \Phi _n^{(1)} (t)\}$ has a finite limiting distribution function $\Phi ^{(1)} (t)$ , which is continuous in $\mathbb{R}^1$ , and singular on the interval 0,1]. We also study the sequence $\{ \Phi _n^{(k)} (t),n = \overline {1,\infty } \}$ for k>1.

Keywords:	critical homeomorphism of a circle distribution function of the entrance time thermodynamic formalism
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