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Newton’s method for solutions of quasi-Bessel differential equations. (English) Zbl 0872.30016

Let \(w\) be a solution of the differential equation \(w''(z)+(1+F(z)) w(z)=0\) where \(F\) is a rational function with a pole at \(\infty\) of order at least 2. Newton’s method of finding the zeros of \(w\) is to iterate the function \(f(z)=z-w(z)/w'(z)\). Denote the \(n\)-th iterate of \(f\) by \(f^{n}\). Suppose that \(f\) is meromorphic in the plane and transcendental. Let \(z_1, z_2,...,z_{N}\) be the zeros of \(1+F\). It follows from a result of Terglane and the reviewer [Trans. Am. Math. Soc. 348, No. 1, 1-12 (1996; Zbl 0842.30021)] that if \(f^{n}(z_{j})\) converges to a finite limit as \(n\rightarrow \infty\) for all \(j\in \{1,2,...,N\}\), then \(f^{n}(z)\) converges to zeros of \(w\) on poles of \(F\) for all \(z\) in the Fatou set of \(f\). The set of non-convergence is in this case thus precisely the Julia set \(J(f)\) of \(f\). The main theorem of the paper under review says that \(J(f)\) has measure zero under the above hypotheses.
This theorem is obtained as a consequence of a quite general result saying that the Julia set of a meromorphic function has measure zero if certain hyperbolicity conditions are satisfied and if it is “thin at infinity”. This extends previous work of C. McMullen [Trans. Am. Math. Soc. 300, 329-342 (1987; Zbl 0618.30027)] and G. M. Stallard [Math. Proc. Camb. Philos. Soc. 108, No. 3, 551-557 (1990; Zbl 0714.30037)]. To deduce the main theorem from this latter result, however, takes still considerable effort. To verify that the Julia set is thin at infinity the author estimates the size of the basins of attraction of the zeros by using results of Hille on the asymptotic behavior of solutions of linear differential equations.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
34M99 Ordinary differential equations in the complex domain
65H05 Numerical computation of solutions to single equations
37F99 Dynamical systems over complex numbers
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