The following elementary identities hold for a cocycle $\mathcal{L}$ over $(\Omega, \sigma, \mu)$, along forward and backward orbits; they are used throughout.

• $\mathcal{L}^{(1)}_\omega = \mathcal{L}_\omega$ and $\mathcal{L}^{(m+n)}_\omega = \mathcal{L}^{(m)}_{\sigma^n\omega} \mathcal{L}^{(n)}_\omega$ (the cocycle law), in particular $\mathcal{L}^{(n+1)}_\omega = \mathcal{L}^{(n)}_{\sigma\omega}\mathcal{L}_\omega$;
• $\sigma^j \sigma^{-(j+n)}\omega = \sigma^{-n}\omega$;
• if $E$ is a family of $m$-dimensional subspaces with $\mathcal{L}_{\sigma^k\omega}E(\sigma^k\omega) = E(\sigma^{k+1}\omega)$ for every $k$, then $\mathcal{L}^{(n)}_\omega E(\omega) = E(\sigma^n\omega)$ for every $n$;
• if $(y_n)$ is a backward history at $\omega$, that is $\mathcal{L}_{\sigma^{-(n+1)}\omega}\, y_{n+1} = y_n$ for all $n$, then $\mathcal{L}^{(n)}_{\sigma^{-n}\omega}\, y_n = y_0$;
• if subspaces $W(\zeta)$ satisfy $\mathcal{L}_{\sigma^{-(m+1)}\omega} W(\sigma^{-(m+1)}\omega) \subseteq W(\sigma^{-m}\omega)$ for all $m$, then $\mathcal{L}^{(n)}_{\sigma^{-n}\omega} W(\sigma^{-n}\omega) \subseteq W(\omega)$;
• an almost-everywhere property holds almost everywhere at every point of the backward orbit.