Let $T = S^1 \times D^2$ be the solid torus, where $D^2 = \{ z \in \mathbb{C} : |z| \le 1\}$ and $S^1 = [0, 1]$ mod $1$. Then we define a map $f : T \to T$, $$f(t, z) = (g(t), \frac{z}{4} + \frac{1}{2} e^{2 \pi t i}),$$ where $g : S^1 \to S^1,\; g(t) = 2 t$ mod $1$ is the doubling map.
I don't understand the geometrical interpretation of $\frac{z}{4} + \frac{1}{2} e^{2 \pi t i}$. Can someone explain me, please?