Let $\mathcal{L}$ be a cocycle on a Banach space $X$ over $(\Omega, \sigma, \mu)$. A backward history of $x \in X$ at $\omega$ is a sequence $y_0 = x, y_1, y_2, \ldots$ in $X$ with

so that $\mathcal{L}^{(n)}_{\sigma^{-n}\omega}\, y_n = x$. For $c \in [-\infty, \infty]$, $B_c(\omega)$ is the set of $x \in X$ having a backward history that decays at rate $c$:

that is, for every real $b < c$, $\|y_n\| \le e^{-bn}$ for all large $n$. In Lean $\sigma^{-n}\omega$ is written $σ.symm^[n] ω$.