The Lyapunov exponents of generic skew-product compact semiflows

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Let $${\mathscr {F}}_K$$ denote the set of infinite-dimensional cocycles over a $$\mu $$ -ergodic flow $$\varphi ^t:M\rightarrow M$$ and with fiber dynamics given by a compact semiflow on a Hilbert space. We prove that there exists a residual subset $${\mathscr {R}}$$ of $${\mathscr {F}}_K$$ such that for $$\Phi \in {\mathscr {R}}$$ and for $$\mu $$ -almost every $$x\in M$$ , either: