norm:

$$\| S^*\| = \max_{b \in \L} \|S^*(b)\| .$$ For any $\delta > 0$ small enough, the metric space $\Ga^*$ with the distance $\rho (S^*_1, S^*_2) = ||S^*_1 - S^*_2||$ is complete. Indeed, if $\b^*_n \to \b^*$ and $\mLip \b^*_n \le D\delta/2 $, then $\mLip \b^* \le D\delta/2$.\
Now consider a map $\G:B\times M \rightarrow B \times M$, like in the setup of Theorem \[thm:mt\]. A central-stable, central-unstable or central lamination is called $\G-$invariant if its leaves $W^*_b$ satisfy: $$\label{eqn:invfol1} \G(W^{s}_{b}) \subset W^s_{h(b)}, \textrm{
}\textrm{ }\textrm{ } W^u_{h(b)} \subset \G(W^u_b)$$ $$\label{eqn:invfol2}\textrm{or }\textrm{ }\textrm{ } W_{h(b)} = \G(W_b).$$ These conditions can all be written in terms of the maps $\beta^*_b$ defining these leaves, and thus in terms of laminations $S^*$ themselves. This will be done in the beginning of Subsection \[sub:graph\].\
Our plan for the proof of Statement $a)$ of Theorem \[thm:mt\] will be the following: we will use the graph transform method described in the following subsection to find $\G-$invariant central-stable and central-unstable laminations. Then the central lamination will be given by $$W_b=W^s_b\cap W^u_b.$$ Property will follow from , so the central lamination will also be invariant under $\G$. Once we have the central lamination, we will define $$Y=\bigsqcup_{b\in \Lambda} 