## 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

### Keywords:

holomorphic dynamics; Siegel disks; analytic functions

### Citations:

Zbl 0263.34003; Zbl 0836.30001
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### References:

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