Let $\mathcal{L}$ be a cocycle on a Banach space $X$ and $\lambda_1 > \lambda_2 > \cdots$ its distinct exponents. For $l \ge 1$, the decomposition after the first $l$ fast spaces (level $l$) consists of measurable families of

• the newest fast space $E(\omega) = E_l(\omega)$, of dimension $m \ge 1$;
• the sum $F(\omega) = E_1(\omega) \oplus \cdots \oplus E_l(\omega)$ of all fast spaces so far, of dimension $M$;
• a slow space $V(\omega) \in \mathcal{G}^M X$ and a projection $Q(\omega)$ onto $V(\omega)$ along $F(\omega)$,

such that, almost surely: $\lambda_l$ is real, $\lambda_{l+1} < \lambda_l$, and $\lambda_{l+1}$ first occurs at index $M + 1$ among the $\chi_k$ (so $M = m_1 + \cdots + m_l$); $E$ is equivariant, has dimension $m = m_l$ and grows at rate $\lambda_l$ in norm and in slowest growth; $E \subseteq F$, and $F$ grows at rate at least $\lambda_l$; $Q$ is strongly measurable, tempered, idempotent with range $V$ and kernel $F$, and commutes with the cocycle; and $\mathcal{L}^{(n)} Q$ grows at rate $\lambda_{l+1}$, with $V(\omega) = \{x : \lambda_\omega(x) \le \lambda_{l+1}\}$.

These are the stages of the induction by which the decomposition is built in the proof of Theorem 3 of Lee (2024).