A leafwise path determines a germ of a transverse diffeomorphism

A finite chart chain exists. The sets $a^{-1}(U)$, over foliation charts, cover $[0,1]$. By [F6] they have a Lebesgue number $\delta >0$. Choose $N$ with $1\mathrm{/}N<\delta$ and put $t_i=i\mathrm{/}N$. Each parameter interval $[t_{i-1},t_i]$ has diameter $1\mathrm{/}N$, so it is contained in some $a^{-1}(U_i)$; equivalently its image is contained in $U_i$. At an intermediate point a small slice $\{x=x(a(t_i))\}$ in a product foliation box is an embedded $q$-dimensional transversal, since its tangent space is complementary to $D$. Thus the required intermediate transversals exist. The subsequent argument applies also to any displayed admissible chain.

Transport inside one chart. Let $p,p^{\mathrm{\prime }}$ lie in one plaque of a foliation chart $U$ with coordinates $(x,y)$, and let $S,S^{\mathrm{\prime }}$ be local transversals at those points. The restriction of $y$ to either transversal has invertible differential: its kernel is the intersection of the transversal tangent space with $D=\ker dy$, which is zero, and its source and target have dimension $q$. By [F4], shrink to neighborhoods $A\subseteq S$ and $B\subseteq S^{\mathrm{\prime }}$ on which these restrictions are diffeomorphisms with the same open image about $y(p)=y(p^{\mathrm{\prime }})$. Then $h=(y{\mathrm{\mid }}_B{)}^{-1}\circ (y{\mathrm{\mid }}_A)$ is a diffeomorphism matching transverse coordinates. It matches points in the same plaque of $U$ after shrinking $A,B$: by [F5], connect $x(p)$ to $x(p^{\mathrm{\prime }})$ by a polygonal path in the open slice at $y(p)$. Its compact image has a finite cover by product boxes contained in the chart image; sufficiently small changes of the transverse value keep this path in those boxes. Short segments in the endpoint boxes connect it to $x(u)$ and $x(h(u))$ for $u$ near $p$. Thus $u$ and $h(u)$ lie in one connected level-set component. If the leaf dimension is zero, the plaque is a singleton and the same conclusion holds in a small transverse box.

The chain composes. To establish the single-plaque assertion, give $L$ the intrinsic structure of [F8]. Each plaque of $U_i$ contained in $L$ is open in $L$: its inclusion factors smoothly through $L$, with invertible differential since both tangent images equal $D$, so [F4] applies. Distinct plaques are disjoint, and an enumerated basis of $L$ assigns to each plaque the least index of a nonempty basic set contained in it. Thus $L$ meets at most countably many plaques of $U_i$ and has at most countably many transverse values there. Each transverse coordinate of the connected continuous image $a([t_{i-1},t_i])$ is constant: two distinct values would force a nondegenerate interval of values by connectedness, contradicting [F9]. The image is therefore a connected subset of one level set and lies in one connected component, hence in one plaque. Step 1.2 gives a transport germ $h_i:(T_{i-1},a(t_{i-1}))\to (T_i,a(t_i))$. Shrink representatives successively so every composite is defined near its source point; then $h_N\circ \cdots \circ h_1$ is a local diffeomorphism near $x$ with value $y$. Its germ is the holonomy germ along the displayed data.

Conclusion. By steps 1.1 and 2.1 every leafwise path with its endpoint transversals and a chosen finite chart chain produces a germ $h_a:(T,x)\to (T^{\mathrm{\prime }},y)$ of a local diffeomorphism, namely the composition of the chart-wise plaque transports. Since a finite chart chain always exists by step 1.1, every leafwise path determines at least one such germ.