W_b.$$ Sending $W_b$ to $b$ defines the desired projection map $p:Y\rightarrow \Lambda$ of . Then the $\G-$invariance of the central lamination is precisely equivalent to the commutativity of diagram . We will follow this plan in the next subsections.

The graph transform map {#sub:graph}
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Here we will deal with the $*=s$ case only, since the $*=u$ case is treated similarly. After that, the central case will be treated as described above. We will introduce first a “pointwise” graph transform map:

$$\g_b:\B^s \longrightarrow \B^s$$ that acts on single leaves, and then a “global” graph transform map:

$$\g:\Ga^s \longrightarrow \Ga^s$$ that acts on entire laminations. In both cases, the geometric idea is the same: start with a map $\beta^s:Q^s \times M \longrightarrow Q^u$ as in . Take the corresponding leaf $W_{h(b)}^s \subset B
\times M$, and take its inverse image under $\G$. The claim is that we obtain a different leaf $\overline{W}_{b}^s \subset B \times M$, corresponding to a map $\overline{\beta}^s: Q^s \times M
\longrightarrow Q^u$. Then we define the graph transform map as: $$\g_b(\beta^s)=\overline{\beta}^s.$$ In other words, the graph transform is implicitly defined by the following relation:

$$\label{eqn:implicitdef}
\{\G^{-1}(\ph_{h(b)}(x_s,\beta^s(x_s,m)),m)\}=\{(\ph_b(x_s,\overline{\beta}^s(x_s,m)),m)\}.$$

We will prove in the appendix that the above correctly defines 