Orthogonal polynomials associated with invariant measures on Julia sets

Translation Invariant Julia Sets

We show that if the Julia set J(f) of a rational function f is invariant under translation by one and infinity is a periodic or preperiodic point for f , then J(f) must either be a line or the Riemann sphere.

Fekete Polynomials and Shapes of Julia Sets

We prove that a nonempty, proper subset S of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if S is bounded and Ĉ \ int(S) is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation i...

The Polynomials Associated With A Julia Set

We prove that, with two exceptions, the set of polynomials with Julia set J has the form {σ pn : n ∈ N , σ ∈ Σ} , where p is one of these polynomials and Σ is the symmetry group of J . The exceptions occur when J is a circle or a straight line segment. Several papers [1, 2, 3, 5] have appeared dealing with the relation between polynomials having the same Julia set J (for notation the reader is ...

On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

On Generating Orthogonal Polynomials for Discrete Measures