Dynamical Complexity and $K$-Theory of $L^p$ Operator Crossed Products

We apply quantitative (or controlled) $K$-theory to prove that a certain $L^p$ assembly map is an isomorphism for $p\in [1,\mathrm{\infty })$ when an action of a countable discrete group $\mathrm{\Gamma }$ on a compact Hausdorff space $X$ has finite dynamical complexity. When $p=2$, this is a model for the Baum-Connes assembly map for $\mathrm{\Gamma }$ with coefficients in $C(X)$, and was shown to be an isomorphism by Guentner, Willett, and Yu.