contraction of the graph transform map. The second way is to revisit the graph transform map and to prove simultaneously the fixed point theorem and the rate of contraction estimate for this map. This is done in the present section.

Laminations {#sub:lam}
-----------

Let $B,h,\Lambda$ be as in Subsection \[sub:perhol\]. In the fibers of the bundles $E^s$ and $E^u$ we have the abstract Riemannian metric, while in the fibers of the trivial bundles $B
\times \rr^k$ and $B \times \rr^l$ we have the standard Euclidean metric. The isomorphism $\varphi^s$ of implies that there exist $k$ linearly independent sections of $E^s$. By applying Gram-Schimdt orthonormalization to these sections, it follows that there exist $k$ orthonormal sections of $E^s$. Sending a fixed orthonormal basis of $\rr^k$ to these orthonormal sections will give us a *metric-preserving* isomorphism $B \times \rr^k
\rightarrow E^s$, and it is this isomorphism that we will henceforth denote by $\varphi^s$. The same discussion applies to $\varphi^u$.\
For any $\delta>0$, we define $Q^s(\delta)$ and $Q^u(\delta)$ to be the open balls of radius $\delta$ around the origin of $\rr^k$ and $\rr^l$, respectively. The metric-preserving isomorphisms $\varphi^s$ and $\varphi^u$ induce metric-preserving isomorphisms in each fiber: $$\label{eqn:trivfiber} \varphi^s_b(\delta):Q^s(\delta) \rightarrow
Q^s_b(\delta), \textrm{ }\textrm{ }\textrm{ 