Self-similarity of Julia sets of the composition of polynomials

Abstract

In the classical iteration theory we say that for a given polynomial $f$ a point $z_0\in\C$ belongs to the Julia set if the sequence of iterates $(f^n)$ is not normal in any neighbourhood of $z_0$. In this paper, we look at the set of non-normality of $(F_n)$, $F_n:=f_n\circ\cdots\circ f_1$, where $(f_n)$ is a given sequence of polynomials of degree at least two. If we can find a hyperbolic domain $M$ which is invariant under all $f_n$, $n\in\N$, $\infty\in M$ and $F_n\to\infty\ (n\to\infty)$ locally uniformly in $M$, then we ask whether these sets of non-normality, which we will also call Julia sets, have properties which we know from the classical case. We show that the Julia set is self-similar. Furthermore, the Julia set is perfect or finite. The finite case may actually occur. We will also give some sufficient conditions for the Julia set being perfect. In the last section we give some examples of sequences of polynomials (where no domain $M$ exists) which have a pathological behaviour in contrast to the classical case.