Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form

If $T=\{T_t\}_{t\in\mathbb R}$ is an aperiodic measure-preserving jointly continuous flow on a compact metric space $\Omega$ endowed with a Borel probability measure $m$, and $G$ is a compact Lie group with Lie algebra $L$, then to each continuous map $A: \Omega \to L$ associate the solution $\Omega\times\mathbb R$ ∋ $(\omega,t)\mapsto X^A(\omega,t)\in G$ of the family of time-dependent initial-value problems $X'(t) = A(T_t\omega)X(t)$, $X(0) =$ identity, $X(t) \in G$ for $\omega\in \Omega$. The corresponding skew-product flow $T^A=\{T_t^A\}_{t\in\mathbb R}$ on $G\times\Omega$ is then defined by letting $T^A_t(g,\omega ) = (X^A(\omega ,t)g,T_t\omega)$ for $(g,\omega)\in G\times\Omega$, $t\in\mathbb R$. The flow $T^A$ is measure-preserving on $(G\times \Omega,\nu_{_G}\otimes m)$ (where $\nu_{_G}$ is normalized Haar measure on $G$) and jointly continuous. For a given closed convex subset $S$ of $L$, we study the set $C_{erg}(\Omega ,S)$ of all continuous maps $A: \Omega\to S$ for which the flow $T^A$ is ergodic. We develop a new technique to determine a necessary and sufficient condition for the set $C_{erg}(\Omega ,S)$ to be residual. Since the dimension of $S$ can be much smaller than that of $L$, our result proves that ergodicity is typical even within very "thin'' classes of cocycles. This covers a number of differential equations arising in mathematical physics, and in particular applies to the widely studied example of the Rabi oscillator.