The set of all holomorphic Euclidean isometries preserving the Julia set of a rational map R is denoted by \(\Sigma R\) . It is shown in this article that if a root-finding method F satisfies the Scaling property (i.e., for a polynomial q, \(F_q\) is affine conjugate to \(F_{\lambda q \circ T}\) for every nonzero complex number \(\lambda \) and every affine map T) then for a centered polynomial p of degree at least two (which is not a monomial), \(\Sigma p\subseteq \Sigma F_p\) . As the Chebyshev’s method satisfies the Scaling property, we have \(\Sigma p \subseteq \Sigma {C_p}\) . Then the article explores the situations where equality holds and in the process, the dynamics of \(C_p\) is found. We show that the Julia set \(\mathcal {J}(C_p)\) of \( C_p\) can never be a line. If a centered polynomial p is (a) having exactly two roots with the same multiplicity, (b) unicritical, (c) cubic and \(\Sigma p\) is non-trivial or (d) quartic, 0 is a root of p and \(\Sigma p \) is non-trivial then it is proved that \(\Sigma p = \Sigma C_p\) . It is found in all these cases that the Fatou set \(\mathcal {F}(C_p)\) is the union of all the attracting basins of \(C_p\) corresponding to the roots of p and the Julia set \(\mathcal {J}(C_p)\) is proved to be locally connected.
