Examples of diffeomorphism group cocycles with no periodic approximation

We construct a finitely generated subgroup of ${\text{Diff}}^{\mathrm{\infty }}({\mathbb{S}}^3\times {\mathbb{S}}^1)$ where every element is conjugate to an isometry but such that the group action itself is far from isometric (the group has “exponential growth of derivatives”). As a corollary, one obtains a locally constant $$\text{Diff}$^{\infty}($\mathbb{S}$^3 \times $\mathbb{S}$^1)$ valued cocycle over a hyperbolic dynamical system which has elliptic behavior over its periodic orbits but which preserves a measure with non-zero top Fiber Lyapunov exponent. Additionally, we provide new examples of Banach cocycles not satisfying the periodic approximation property as first shown by Kalinin-Sadovskaya.