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

If T = {T_t }_{t ∈ R} is an aperiodic measure-preserving jointly continuous flow on a compact metric space Ω endowed with a Borel probability measure m, and G is a compact Lie group with Lie algebra L, then to each continuous map A: Ω → L associate the solution Ω×R ∋ (ω, t) → X^A (ω, t) ∈ G of the family of time-dependent initial-value problems X^′ (t) = A(T_t ω)X (t), X (0) = identity, X (t) ∈ G for ω ∈ Ω. The corresponding skew-product flow T^A = {T_t ^A }_{t ∈ R} on G× Ω is then defined by letting T_t ^A (g,ω) = (X^A (ω, t)g,T_t ω) for (g,ω)∈ G×Ω, t ∈ R. The flow T^A is measure-preserving on (G×Ω,ν_G ⊗ m) (where ν_G is normalized Haar measure on G) and jointly continuous. For a given closed convex subset S of L, we study the set C_{er g} (Ω,S) of all continuous maps A: Ω→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_{er g} (Ω,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.