In this article, we study the global dynamics of Halley’s method applied to complex polynomials. Specifically, we analyze the structure and connectedness of the Julia set of this method. The convergence behavior, symmetry properties, and topological features of the corresponding Fatou and Julia sets are studied for various classes of polynomials, including unicritical, cubic, and quartic polynomials with non-trivial symmetry groups. In particular, we prove that Halley’s method <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> is convergent, its Julia set is connected, the immediate basins are unbounded, and the symmetry group of it coincides with that of the polynomial whenever p belongs to one of the above classes. We further extend our results to polynomials with roots satisfying rotational symmetry, namely <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p(z)=z(z^n+a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>z</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mi>n</mi> </msup> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a\in \mathbb {C}\setminus \{0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. It is shown that the immediate basin of Halley’s method <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> corresponding to a root of p can be bounded when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\ge 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>. We also make some remarks on the dynamics of Halley’s method applied to a cubic polynomial in general.

On the dynamics of Halley’s method

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发表日期：

2026年2月26日

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Gang Liu,
Soumen Pal,
Saminathan Ponnusamy

摘要

In this article, we study the global dynamics of Halley’s method applied to complex polynomials. Specifically, we analyze the structure and connectedness of the Julia set of this method. The convergence behavior, symmetry properties, and topological features of the corresponding Fatou and Julia sets are studied for various classes of polynomials, including unicritical, cubic, and quartic polynomials with non-trivial symmetry groups. In particular, we prove that Halley’s method $H_p$ $H_{p}$ is convergent, its Julia set is connected, the immediate basins are unbounded, and the symmetry group of it coincides with that of the polynomial whenever p belongs to one of the above classes. We further extend our results to polynomials with roots satisfying rotational symmetry, namely $p(z)=z(z^n+a)$ $p (z) = z (z^{n} + a)$ , where $a\in \mathbb {C}\setminus \{0\}$ $a \in C \ {0}$ . It is shown that the immediate basin of Halley’s method $H_p$ $H_{p}$ corresponding to a root of p can be bounded when $n\ge 7$ $n \geq 7$ . We also make some remarks on the dynamics of Halley’s method applied to a cubic polynomial in general.