Let $S$ be a vertical strip-like domain in $\mathbb{C}$ bounded by two simple curves. $S$ is periodic. That means that there is a real constant $c>0$, such that $S = S + ic$ and also $S = S + ic\cdot k$ for every $k \in \mathbb{Z}$. Lets cut $S$ with horizontal cuts into congruent parts which are exactly $c$ tall and are from now on called $S_k$. $S_k$ can be mapped one-to-one and holomorphic onto a rectangle with width $1$ and a specific height $r$. Lets call this function $f_k$. We map every $S_k$ onto such a rectangle $R_k$ in a way that $$R_k = \lbrace z \in \mathbb{C} | 0 < \Re(z) < 1 \text{ and } r\cdot k < \Im(z) < r\cdot (k+1) \rbrace$$ for every $k \in \mathbb{Z}$. To win a holomorphic map from $S$ onto the vertical strip $$T = \lbrace z \in \mathbb{C} | 0 < \Re(z) < 1 \rbrace$$ we need to complete the mapping by adding the missing cuts. Since each $S_k$ is bounded by two cuts - which are basically horizontal lines - the mapping has an analytic continuation. So $f_k$ can also be defined on one of the cuts.This should complete the mapping from $S$ onto $T$. The question is:
How can I am sure that I can assemble those continued mappings to a holomorphic map from $S$ onto $T$ such that $T$ is also periodic? It must come from the periodicity.