Dynamical classification of analytic one-frequency quasi-periodic $\mathrm{S}\mathrm{O}(3,\mathbb{R})$-cocycles

We establish a close connection between acceleration and dynamical degree for one-frequency quasi-periodic compact cocycles, by showing that two vectors derived separately from each coincide. Based on this, we provide a dynamical classification of one-frequency quasi-periodic $\mathrm{S}\mathrm{O}(3,\mathbb{R})$-cocycles.