Polynomials without repelling periodic point of given period
Let \(P\) be a polynomial and denote by \(P^k\) the \(k\)-th iterate of \(P.\) A point \(z_0\) is called a periodic point of period \(k\) if \(f^k(z_0)=z_0\) but \(f^j(z_0)\neq z_0\) for \(j<k.\) A periodic point of period \(k\) is called repelling if \(| (f^k)'(z_0)| >1.\) It was shown by the reviewer [Habilitationsschrift, RWTH Aachen (1991)] that if there exists a polynomial of degree \(d\geq 2\) which has no periodic point of period \(k\geq2,\) then \((k,d)\in \{(2,2),(2,3), (2,4),(3,2)\}.\) Moreover, it was shown that exceptional polynomials do indeed occur in each of the four cases. In the present paper all exceptional polynomials are determined; that is, the polynomials for which some iterate fails to have repelling periodic points of some period are classified. It is also shown that the Julia sets of these polynomials are connected.
scientific article; zbMATH DE number 5078042