Disconnected Julia set of Halley’s method for exponential maps. (English) Zbl 1501.37044
Authors’ abstract: We investigate the Halley method of exponential maps. Our main result is that, unlike Newton’s method, the Julia set of Halley’s method may be disconnected when applied to entire maps of form \(F(z)=p(z)e^{q(z)}\) where \(p(z)\) and \(q(z)\) are polynomials and \(q(z)\) is non-constant. We also describe the nature of the fixed points and classify rational Halley’s maps of entire functions.
Reviewer: Haifeng Chu (Xi’an)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
| 30D20 | Entire functions of one complex variable (general theory) |
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