will often write simply $Q^s,Q^u,Q^s_b,Q^u_b,B_b,\ph^s_b,\ph^u_b,\ph_b,h_b,(h^{-1})_b$ for the notions introduced in the previous paragraphs. By and the fact that are metric-preserving, $dh_b$ has block-diagonal form at $0$: $$dh_b(0) = \textrm{diag} (A^u, A^s),$$ where $\lambda_-\leq |A_s|\leq \lambda$ and $\mu_-\leq
|A_u^{-1}|\leq \mu$. Because the coordinate charts $\ph_b$ are smooth functions, we have the following estimate throughout $Q^s\times Q^u$: $$\label{eqn:esth} {||dh_b -
\textrm{diag} (A^u, A^s)||}_{C^0} \le O(\delta ).$$

Now consider another compact manifold $M$, as in the statement of Theorem \[thm:mt\]. For any domain $A$ and any mapping $\b : A \to B$, we will denote by $\ga (\b )$ the map from $A$ onto the graph:

$$\label{eqn:gamma} \ga(\b):A \rightarrow A \times
B, \textrm{ }\textrm{ }\textrm{ } \ga (\b ): a \mapsto (a, \b (a))
\in A \times B.$$

Statement $a)$ of Theorem \[thm:mt\] provides a correspondence between leaves and base points, so it’s about time we defined these. The leaves of center-stable, center-unstable and center foliations corresponding to $b \in \L$ are represented by Lipschitz maps: $$\label{eqn:leaf} \beta^s_b:Q^s\times M \rightarrow Q^u,\textrm{
}\textrm{ } \textrm{ }\beta^u_b:Q^u\times M \rightarrow Q^s,\textrm{
}\textrm{ } \textrm{ }\beta_b:M \rightarrow Q^u \times Q^s.$$ Then we define the leaves to be simply the graphs of the Lipschitz maps, embedded in $B\times 