$\overline{\beta}^s$ (in other words, that the Implicit Function Theorem applies). The above definition also works in families. For a lamination $S^s \in \Gamma^s$ with leaves that are graphs of $\beta^s=S^s(h(b))$, define its graph transform as: $$\g(S^s)=(\overline{S}^s),$$ where $\overline{S}^s(b)=\overline{\beta}^s$ is defined by relation .\
Comparing with , we see that a lamination $S^s$ is $\G-$invariant if and only if it is a fixed point of the graph transform map $\g$. Therefore, to show that there exists a unique $\G-$invariant central-stable lamination, we will use the fixed point principle: it is enough to show that $\g$ is well defined and contracting.

\[lem:1\] For $\rho$ small enough and any $\F, \G$ as in Theorem \[thm:mt\], there exists $\delta=O(\rho)$ so that the graph transform $\g$ maps $\Ga^s$ into itself and is contracting with Lipschitz constant $\mu+O(\delta)$. In other words, for any $S^s_0,
S^s_1 \in \Ga^s $ we have: $$\label{eqn:cont1} \|
\g(S_0^s) - \g(S_1^s) \| \le (\mu + O(\delta))\| S_0^s - S_1^s\| .$$ In the pointwise situation, for any $b\in \Lambda$, we claim that $\g_b$ maps $\B^s$ into itself. Furthermore, for any $\b^s_0,\beta^s_1 \in \B^s$, we have: $$\label{eqn:36a}
\| \g_b(\b_0^s) - \g_b(\b^s_1)\| \leq (\mu + O(\delta ))\|  {\b^s_0} - {\b^s_1} \|.$$

\[cor:fixed\] For $\delta=O(\rho)$ small 