# $\mathbb{R}$-action on the unit disk.

I want to prove that there exists an $\mathbb{R}$-action on the unit disk $D$ in complex plane which preserves $i$ and $-i$, where $D=\{z\in \mathbb{C}~|~|z|\leq 1\}$.

This arises when we define the boundary map of Heegaard Floer homology and not a homework.

I know that one way to prove this is to find a conformal map from the strip $S$ to the disk $D$, where $S=\{z\in \mathbb{C}~|~0<\operatorname{Re} z<1\}$ and use $\mathbb{R}$-action on $S$.

However, I don't know any specific conformal map from $S$ to $D$.

Any ideas?

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You want a large set of conformal maps $\hat T:\ D\to D$ keeping the two points $\pm i$ fixed. "Up to a conjugation" this is the same thing as a large set of conformal maps $T$ mapping the upper half plane $H$ (ergo: the real axis) to itself and keeping the two points $0$ and $\infty$ fixed. The maps $$T_\lambda: \quad \bar {\mathbb C}\to \bar {\mathbb C}\,\qquad z\mapsto e^\lambda \ z\qquad(\lambda\in{\mathbb R})$$ obviously have this property, and in addition we have $T_{\lambda+\mu}=T_\lambda\circ T_\mu\$.
Now conjugate the $T_\lambda$ with a conformal map that maps $H$ onto $D$ and $0$, $\infty$ to $i$, $-i$.