A cocycle consists of an invertible ergodic measure-preserving transformation $\sigma$ of a Lebesgue probability space $(\Omega, \mu)$ (a standard Borel space with a probability measure giving points measure zero), together with a map $\omega \mapsto \mathcal{L}_\omega$ from $\Omega$ into the bounded operators on a real Banach space $X$ which is

• strongly measurable: $\omega \mapsto \mathcal{L}_\omega x$ is measurable for each $x \in X$;
• forward integrable: $\log^+ \|\mathcal{L}_\omega\| \in L^1(\mu)$.

No invertibility of $\mathcal{L}_\omega$ is assumed. Its iterates are $\mathcal{L}^{(0)}_\omega = \mathrm{id}$ and $\mathcal{L}^{(n+1)}_\omega = \mathcal{L}_{\sigma^n \omega} \circ \mathcal{L}^{(n)}_\omega$.

All logarithms below take values in $[-\infty, \infty)$ with $\log 0 = -\infty$, and indices of exponents are shifted by one from the usual convention: $lambda 0$ is $\lambda_1$.

• The growth rate of a vector is $\lambda_\omega(x) = \limsup_n \frac1n \log \|\mathcal{L}^{(n)}_\omega x\|$.
• The $k$-th Lyapunov exponent ($k \ge 1$) is $\chi_k(\omega) = \limsup_n \frac1n \log \rho_k(\mathcal{L}^{(n)}_\omega)$, with $\rho_k$ the Bernstein number. The sequence $\chi_1 \ge \chi_2 \ge \cdots$ is non-increasing and $\chi_1 = \lambda_1$ is the top exponent.
• The distinct exponents $\lambda_1 > \lambda_2 > \cdots$ are the distinct values of $\chi_k$: $\lambda_1 = \chi_1$ and $\lambda_{i+1} = \chi_t$ for the least $t$ with $\chi_t < \lambda_i$. The multiplicity $m_i$ of $\lambda_i$ is the number of $k$ with $\chi_k = \lambda_i$. When there is no exponent after $\lambda_i$ the recursion stops: $m_i$ is recorded as $0$, and the later $\lambda$'s are $-\infty$.
• The index of compactness is $\nu(\omega) = \lim_k \chi_k(\omega) = \inf_k \chi_k(\omega)$.
• The cocycle is quasicompact if $\nu < \lambda_1$ almost everywhere.
• A family of operators $\omega \mapsto P_\omega$ is tempered if $\omega \mapsto \log \|P_\omega\|$ is tempered for $(\sigma, \mu)$.