A periodic C1 characteristic map is an orientation-preserving diffeomorphism

By [F1], both the forward flow ${\mathrm{\Phi }}_{t,s}$ and the reverse-time flow ${\mathrm{\Phi }}_{s,t}$ are defined. Uniqueness of the ODE implies that following the same characteristic first from $s$ to $t$ and then backward to $s$ returns the initial point; likewise in the opposite order. Thus ${\mathrm{\Phi }}_{s,t}\circ {\mathrm{\Phi }}_{t,s}=\mathrm{i}\mathrm{d}$ and ${\mathrm{\Phi }}_{t,s}\circ {\mathrm{\Phi }}_{s,t}=\mathrm{i}\mathrm{d}$. By [F2] both maps are $C^1$, so ${\mathrm{\Phi }}_{t,s}$ is a $C^1$ diffeomorphism.

Applying [F3] to ${\mathrm{\Phi }}_{s,t}\circ {\mathrm{\Phi }}_{t,s}=\mathrm{i}\mathrm{d}$ gives $D{\mathrm{\Phi }}_{s,t}({\mathrm{\Phi }}_{t,s}(x))D{\mathrm{\Phi }}_{t,s}(x)=I_d$. Hence $D{\mathrm{\Phi }}_{t,s}(x)$ is invertible, and [F4] gives $\det D{\mathrm{\Phi }}_{t,s}(x)\mathrm{\ne }0$.

For fixed $s,x$, the function $r\mapsto \det D{\mathrm{\Phi }}_{r,s}(x)$ is continuous by [F2] and [F5], and equals $1$ at $r=s$. It never vanishes by step 2.1. If it were negative at some $r$, [F6] on the interval between $s$ and $r$ would force a zero, a contradiction. Therefore it is positive at every $r$, including the chosen $t$.