space, $d\gamma^F$, associated with a collection of forward trajectories, is taken to be the phase space volume $d\Gamma_0^F$ occupied by their initial conditions. Liouville’s theorem then implies that the volume occupied by a given set of forward trajectories is equal to that of the conjugate set of reverse trajectories. In this sense, the numerator and denominator in Eq.\[eq:ratio\] are defined with respect to the same measure on trajectory space.)

![ The upper box represents the space of all forward trajectories, the lower box the space of reverse trajectories, and in this visual depiction conjugate pairing is indicated by reflection about a horizontal line between the two boxes (e.g. the two crosses represent a pair of conjugate twins). The grey regions $\zeta_{\rm typ}^F$ and $\zeta_{\rm typ}^R$ denote the peaks of the probability distributions ${\cal P}^F$ and ${\cal P}^R$, while the vertically striped regions $\zeta_{\rm dom}^F$ and $\zeta_{\rm dom}^R$ are the peaks of the functions ${\cal Q}^F$ and ${\cal Q}^R$. The dashed arrows indicate the conjugate pairing that is the central result of this paper (Eq. \[eq:central\]). []{data-label="fig:pathSpace"}](Figure5.eps)

Let us now write $\langle e^{-\beta W_d}\rangle_F$ as an integral over forward trajectories: $$\label{eq:exp_F.q}
1 = \Bigl\langle e^{-\beta W_d}\Bigr\rangle_F = 
\int d\gamma^F\, {\cal P}^F[\gamma^F] 