second factor in the right hand side is $\leq ||{\b^s_0} -{\b^s_1}||$. As for the first factor, we see that: $$\textrm{Lip }(\pi_u \circ \G_{b}^{-1}) \leq
\textrm{Lip }(\pi_u \circ \F_{b}^{-1}) + O(\rho)=\textrm{Lip }(\pi_u
\circ h_{b}^{-1}) + O(\rho) \leq \mu+O(\rho).$$ Since $\rho=O(\delta)$, we conclude that:

$$\label{eqn:t1}
T_1 \leq (\mu+O(\delta))\cdot ||{\b^s_0} -{\b^s_1}||.$$

As for $T_2$, we see that:

$$T_2\leq \textrm{Lip }(\pi_u \circ \G_{b}^{-1} \circ
\ga({\b^s_1}))\cdot ||G_{\overline\b_0,b} - G_{\overline\b_1,b}||.$$ In , we saw that: $$\mLip (\pi_u \circ
\G_{b}^{-1} \circ \gamma({\b^s_1})) \leq O(\delta).$$ In Proposition \[prop:last\] of the Appendix, we will prove that $$||G_{\overline\b_0,b} - G_{\overline\b_1,b}|| \leq O(1)\cdot
||\overline\beta^s_0-\overline\beta^s_1||.$$ Therefore, we obtain:

$$T_2 \leq O(\delta)\cdot ||\overline\beta^s_0-\overline\beta^s_1||.$$ Together with , this implies: $$||\overline\b^s_0-\overline\b^s_1|| \leq (\mu+O(\delta))\cdot ||{\b^s_0} -{\b^s_1}|| + O(\delta)
\cdot ||\overline\beta^s_0-\overline\beta^s_1|| \Rightarrow$$ $$||\overline\b^s_0-\overline\b^s_1|| \leq (\mu+O(\delta))\cdot ||{\b^s_0} -{\b^s_1}||.$$ This is precisely the desired inequality .

The central lamination {#sub:central}
----------------------

Corollary \[cor:fixed\] tells us that there exists a unique $\G-$invariant central stable lamination $S^s \in \Ga^s$. This can be presented either via the maps $\beta^s_b$, or via the leaves $W^s_b$ (as $b$ ranges over $\Lambda$). Similarly, there exists a unique $\G-$invariant central unstable lamination $S^u\in \Ga^u$. Let us define the central lamination $S$ via its leaves $W_b$, which we define by: $$\label{eqn:defcentral}
W_b=W^s_b\cap W^u_b.$$ This lamination will be 