2024/11/18 by Subhasis Ghora, Tarakanta Nayak, Ghora, Subhasis +5 · 1 citation
Engineering · Mathematics · #37F10 #65H05 #Advanced Numerical Analysis Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #History and Theory of Mathematics #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2411.11290
openalex publication_date 2024/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is proved that the Chebyshev's method applied to an entire function f is a rational map if and only if f(z) = p(z) eq(z), for some polynomials p and q. These are referred to as rational Chebyshev maps, and their fixed points are discussed in this article. It is seen that ∞ is a parabolic fixed point with multiplicity one bigger than the degree of q. Considering q(z)=p(z)n+c, where p is a linear polynomial, n ∈ ℕ and c is a non-zero constant, we show that the Chebyshev's method applied to peq is affine conjugate to that applied to z ezn. We denote this by Cn. All the finite extraneous fixed points of Cn are shown to be repelling. The Julia set J(Cn) of Cn is found to be preserved under rotations of order n about the origin. For each n, the immediate basin of 0 is proved to be simply connected. For all n ≤ 16, we prove that J(Cn) is connected. The Newton's method applied to zezn is found to be conjugate to a polynomial, and its dynamics is also completely determined.