MathDB

Let

D\subseteq \mathbb{C}

be a compact set with at least two elements and consider the space \Omega=\bigtimes_{i=1}^{\infty} D with the product topology. For any sequence

(d_n)_{n=0}^{\infty} \in \Omega

let

f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n

, and for each point

\zeta \in \mathbb{C}

with

|\zeta|=1

we define

S=S(\zeta,(d_n))

to be the set of complex numbers

w

for which there exists a sequence

(z_k)

such that

|z_k|<1

z_k \to \zeta

, and

f_{d_n}(z_k) \to w

. Prove that on a residual set of

\Omega

, the set

S

does not depend on the choice of

\zeta

Problem Statement