Taking a similar approach as Leo; this is largely Leo's discussion; but we're going to switch to \( f(z) = z - 1/z \); the reason for this switch is to reduce the group structure.
Let's talk about the abel functions of \( f \); which produce two branches:
\(
\alpha_+(f(z)) = \alpha_+(z) + 1\,\,\text{for}\,\, \Im(z) > 0\\
\alpha_-(f(z)) = \alpha_-(z) + 1 \,\,\text{for}\,\, \Im(z) <0\\
\)
In which:
\(
\alpha_{+}(z) : \mathbb{H} \to \mathbb{H}\\
\alpha_{+}(\infty) = \infty\\
\)
Where:
\(
\mathbb{H} = \{z \in \mathbb{C}\,|\, \Im(z) > 0\}\\
\)
And vice versa with \( \alpha_{-} \). We can define two half iterations as:
\(
g_{12}(z) = \alpha_{\pm}^{-1}(\alpha_{\pm}(z) + 1/2)\\
\)
The beauty of this example is that the domains of definition do not intersect. We have a natural boundary along \( \mathb{R} \) which is the julia set of \( g \). If we take \( g_1(g_2) \) it is nonsense because one exists in a projection to the upper half plane; and the other, the lower half plane.
You are doing something like this, Leo. You're just doing it for a much more complicated group structure. But this is where the fundamental group comes in.
The fundamental group of the above example is just two non intersecting simply connected domains. In your example it's definitely trickier.
I enjoy your posts, Leo; I hope this makes sense.
Let's talk about the abel functions of \( f \); which produce two branches:
\(
\alpha_+(f(z)) = \alpha_+(z) + 1\,\,\text{for}\,\, \Im(z) > 0\\
\alpha_-(f(z)) = \alpha_-(z) + 1 \,\,\text{for}\,\, \Im(z) <0\\
\)
In which:
\(
\alpha_{+}(z) : \mathbb{H} \to \mathbb{H}\\
\alpha_{+}(\infty) = \infty\\
\)
Where:
\(
\mathbb{H} = \{z \in \mathbb{C}\,|\, \Im(z) > 0\}\\
\)
And vice versa with \( \alpha_{-} \). We can define two half iterations as:
\(
g_{12}(z) = \alpha_{\pm}^{-1}(\alpha_{\pm}(z) + 1/2)\\
\)
The beauty of this example is that the domains of definition do not intersect. We have a natural boundary along \( \mathb{R} \) which is the julia set of \( g \). If we take \( g_1(g_2) \) it is nonsense because one exists in a projection to the upper half plane; and the other, the lower half plane.
You are doing something like this, Leo. You're just doing it for a much more complicated group structure. But this is where the fundamental group comes in.
The fundamental group of the above example is just two non intersecting simply connected domains. In your example it's definitely trickier.
I enjoy your posts, Leo; I hope this makes sense.