} \varphi^u_b(\delta):Q^u(\delta) \rightarrow
Q^u_b(\delta),$$ where $Q^s_b(\delta) \subset E^s$ and $Q^u_b(\delta) \subset
E^u$ are the open balls of radius $\delta$ around the origin in the respective fibers.\
The number $\delta$ must be chosen small enough such that for any $b\in B$, the exponential map gives us an open embedding $Q^s_b(\delta) \times Q^u_b(\delta)
\hookrightarrow B$. We write $B_b(\delta)$ for the image of this map. Composing this embedding with the isomorphism $\varphi^s_b(\delta) \times
\varphi^u_b(\delta)$ gives us an open embedding (coordinate chart):

$$\label{eqn:fixiso} \varphi_b(\delta):Q^s(\delta) \times Q^u(\delta) \hookrightarrow B.$$

Let us take $C>\textrm{max}(\lambda_-^{-1},\mu_-^{-1})$, and consider the above constructions for radius $C\delta$. Then we can express the map $h:B\rightarrow B$ locally around $b$ in the domain and around $h(b)$ in the target. Therefore, in coordinates given by the chart , the map $h$ has the form:

$$\label{eqn:hcoord} h_b(\delta) =
(\ph_{h(b)}(C\delta))^{-1} \circ h \circ \ph_b(\delta), \textrm{
}\textrm{ }\textrm{ } {(h^{-1})}_b(\delta) = (\ph_{b}(C\delta))^{-1}
\circ h^{-1} \circ \ph_{h(b)}(\delta).$$

For various values of $\delta$, the maps $h_b(\delta)$ will represent the same germ at $0$, but will have different domains. Similarly, the maps ${(h_b)}^{-1}(\delta)$ and ${(h^{-1})}_b(\delta)$ are representatives of the same germ at $0$, but have different domains.\
Until Section \[sec:hold\], we will work with a single, fixed $\delta$. Therefore, we 