Remark on functions having the same Julia set (Q5930230)

This paper under review is concerned with rational functions with the same Julia set. Throughout the paper \(f\), \(g\) denote nontrivial rational functions in the extended complex plane \(\overline{\mathbb{C}}\). Functions \(f\), \(g\) are called permutable if \(f\circ g=g\circ f\). G. Julia showed in 1923 that for permutable \(f\), \(g\) one has the same Julia set \(J(f)= J(g)\). NEWLINENEWLINENEWLINEOne may ask if a converse of Julia's result holds. For the polynomial case, this problem is settled: if \(f\), \(g\) are polynomials such that \(J(f)= J(g)= J\), then either \(J\) has a rotational symmetry or \(f\), \(g\) are permutable. For the rational case, the converse need not hold. NEWLINENEWLINENEWLINEIn the cases when the Julia set is neither \(\overline{\mathbb{C}}\) nor a part of a circle or of straight line, the class of rational functions \(f\), \(g\) with \(J(f)= J(g)\) should be restricted even that \(f\), \(g\) are permutable. I. Baker and A. Eremenko showed in 1987 that if the Julia set \(J(f)\) has infinitely many cusps then the set of all rational \(g\) is such that \(J(f)= J(g)\) is countably infinite. NEWLINENEWLINENEWLINEThe main result of this paper is as follows: NEWLINENEWLINENEWLINETheorem 1. Let \(f\), \(g\) be nontrivial rational functions having the same Julia set \(J(f)= J(g)\). If \(f\) has a rational indifferent periodic point and that the critical set of \(f\) is disjoint of \(J(f)\). Then either \(J(f)\) has to be equal to a circle, an arc of a circle for some coordinate, or \(f\), \(g\) have to verify an equation of the type associated with \(m_1,\dots, m_k\in \mathbb{N}\) and \(m\in \mathbb{N}^+\): \(f^{m_1}\circ g\circ f^{m_2}\circ g\circ\cdots\circ f^{m_k}\circ g= f^m\).