GEOMETRIC PROPERTIES OF JULIA SETS OF THE COMPOSITION OF POLYNOMIALS OF THE FORM z2 + cn

For a sequence (cn) of complex numbers we consider the quadratic polynomials fcn(z): = z 2 + cn and the sequence (Fn) of iterates Fn: = fcn ◦ · · · ◦ fc1. The Fatou set F(cn) is by definition the set of all z ∈ C ̂ such that (Fn) is normal in some neighbourhood of z, while the complement of F(cn) is called the Julia set J(cn). The aim of this article is to study geometric properties, Lebesgue measure and Hausdorff dimension of the Julia set J(cn) provided that the sequence (cn) is bounded. 1. Introduction. For a sequence (cn) of complex numbers we consider the quadratic polyno-mials fcn(z): = z2+ cn and the sequence (Fn) of iterates Fn: = fcn ◦ · · · ◦fc1. (Note that Fn depends on c1,..., cn which we do not indicate explicitly in the notation.) If cn = c for all n, we write fnc instead of Fn. The Fato