Chebyshev polynomials on generalized Julia sets (Q312713)

For a sequence of polynomials one can easily generalize the notion of the Julia set by taking compositions \(F_m\) of the initial \(m\) elements of the sequence and looking for the set of non-normality for the family \((F_m)_{m=1}^{\infty}\). The classical, or autonomous, Julia set of a single mapping is then obtained from the constant sequence. It is not a well-known but interesting fact that for a monic polynomial \(f\) its iterates have an extremal property with respect to the Julia set \(J(f)\). Namely, each iterate \(f^{\circ n}\) with some constant added to it minimizes the sup-norm on \(J(f)\) among all monic polynomials of its degree. The present paper extends that result to quite general sequences of polynomials. The compositions \(F_m\) then play the role of iterates. They do not have to be monic, since one can normalize \(\hat{F}_m := \rho_m^{-1}F_{m}-\tau_m\) and the claim is again that for some constants \(\rho_m\) and \(\tau_m\) the polynomial \(\hat{F}_m\) is a monic minimizer for its degree of the sup-norm on the Julia set of the sequence.