On the multiplier of a repelling fixed point
The author estimates the multiplier \(\lambda\) of a repelling fixed point with rational rotation number for a polynomial in one complex variable. It is shown that \(\lambda\) lies in a small disk tangent to the imaginary axis. The radius of the disk is determined by the degree of the polynomial, the rotation number of the fixed point and the escape rates and external arguments of the critical values. Theorem 2.8. Let \(f: \mathbb{C}\to \mathbb{C}\), be a degree \(d\) polynomial with \(d\geq 2\). Let \(z_0\) be a repelling fixed point with multiplier \(\lambda\). Suppose that the type \(T\) of \(z_0\) has rotation number \(p/q\) and cardinality \(kq\). Let \(\beta(f, T)\) be the space constant of \(f\) and \(T\). There is a complex number \(L\) for which \(\exp(L+ 2\pi i\cdot p/q)= \lambda\) and \(|L|^2\cdot \beta(f, T)\). This result generalizes an equality proved by J. C. Yoccoz for polynomials whose filled Julia sets are connected [\textit{J. C. Yoccoz}, Sur la taille des membres de l'ensemble de Mandelbrot (preprint)]. Let \({\mathcal E}_d\) be the escape locus consisting of those degree \(d\) polynomials whose critical values all lie in the set of point \(\Omega\) whose orbits are unbounded, the escape rate of \(f\), \(\rho(f)\), to be the maximum of the values \(e(v)\), as \(v\in {\mathcal V}\) ranges over the critical values \(f\) in \(\Omega\). It is given an estimate for the multiplier of a fixed point that is uniform over bounded regions in \({\mathcal E}_d\). Theorem 3.1. Let \(f\in {\mathcal E}_d\) and \(\rho(f)= r\). Let \(z_0\) be a repelling fixed point of \(f\) with multiplier \(\lambda\). Suppose that the type \(T\) of \(z_0\) has rotation number \(p/q\) and cardinality \(kq\). Let \(B(r, d, q)= \arctan 1/r(d^q- 1)\). Then there is a complex number \(L\) such that \(\exp(L+ 2\pi i\cdot p/q)= \lambda\) and \(|L|^2/\text{Re } L\leq 2\pi\cdot \log d/kq\cdot B(r, d, q)\). Finally, the Bers inequality for a quasi-Fuchsian manifold \(M\) is discussed [\textit{L. Bers}, On the boundaries of Teichmüller spaces and on Kleinian groups I. Ann. Math., II. Ser. 91, 570-600 (1970; Zbl 0197.06001)]. This inequality relates the Poincaré lengths of geodesics in the boundary of \(M\) to the hyperbolic length of corresponding geodesics in the interior. The Yoccoz and Bers inequalities are analogous both in statement and proof; they constitute an entry in the conceptual dictionary between rational maps and Kleinian groups.
scientific article; zbMATH DE number 703877
scientific article