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Farey curves. (English) Zbl 1035.37033

Let \(P_\lambda:z\to\lambda z+z^2\) be a quadratic polynomial, where \(| \lambda| \leq1\) be a complex number. The unique analytic function \(\varphi_\lambda\) of the form \(\varphi_\lambda(z)=z+ \mathcal{O} (| z| ^2)\) that linearizes \(P_\lambda\) on the unit disk \(\mathbb{D}= \{| z| <1\}\) is defined by the formula \(\varphi_\lambda(z)= \lim_{n\to+\infty}P^{\,\circ\,n}_\lambda(z)\lambda^n\). (In other words, under the change of coordinates, \(\xi=\varphi_\lambda(z)\), the expression of \(P_\lambda\) simply becomes the multiplication by \(\lambda\)).
The authors study the bounded analytic function \(\eta:\mathbb{D}\to\mathbb{C}\) by \(\eta(\lambda)= \varphi_\lambda (-\lambda/ 2)\), noting that \(\eta\) has radial limits almost everywhere on \(\partial \mathbb{D}=\{| \lambda| =1\}\) [see also A. D. Bryuno, Tr. Mosk. Mat. Obsc. 25, 119-262 (1971; Zbl 0263.34003), and J.-C. Yoccoz, Astérisque. 231. Paris: Socièté Math. de France (1995; Zbl 0836.30001)]. In particular, the behavior of \(\eta(\lambda)\) is illustrated on concentric circles \(| \lambda| \in\{.5,\,.75,\,.9,\,.99,\,.999,\,.9999\}\) by series of figures with the so-called Farey curves \(\eta(| \lambda| \exp(2\pi it))\).

MSC:

37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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References:

[1] Brjuno A. D., Trudy Moskov. Mat. Obšč. 25 pp 119– (1971)
[2] Cremer H., Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 84 (1932)
[3] Duren P. L., Univalent functions (1983)
[4] Fatou P., C. R. Acad. Sci. Paris 143 pp 546– (1906)
[5] DOI: 10.1007/BF01443992 · JFM 02.0200.01
[6] DOI: 10.2307/1968952 · Zbl 0061.14904
[7] Yoccoz, J. C. 1995.Petits diviseurs en dimension1France: Soc. math. [Yoccoz 1995], Astérisque 231
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