$n$ Torus contained in the closure of the image of the unit disc under a holomorphic map?

Question

I have the following question. Does there exists a holomorphic function $\varphi\in\mathcal{O}(\mathbb{D},\mathbb{D}^{n})$ such that $\mathbb{T}^n\subseteq\overline{\varphi(\mathbb{D})},$ where $n\geq2$?

Answer 1

This is true for $n=2$. I'll use a result of C. L. Belna, P. Colwell and G. Piranian from The radial behavior of Blaschke products, a special case of which reads as: there exists a holomorphic function $f:\mathbb D\to\mathbb D$ and a dense subset $A\subset \mathbb T$ such that for every $a\in A$, the cluster set of $f$ at $a$ is $\mathbb T$. (In the notation of their theorem, make $\{\zeta_m\}$ dense in $\mathbb T$ and let $K_m=\mathbb T$ for all $m$.)

For $n=2$, the map $\varphi(z)=(z,f(z))$ has the desired property. Indeed, by the choice of $f$ the closure of $\varphi(\mathbb D)$ contains $A\times \mathbb T$, which is dense in $\mathbb T^2$.

I think the answer should be affirmative for general $n$, but my attempt with $\varphi(z)=(z,f(z),f(f(z)))$ was not conclusive.