Two representation theorems are presented:

1. Any continuous action of $\mathbb R^k$ ($k\in \mathbb N$) on a metrizable compact space $X$ admits an injective $G$-equivariant continuous map into $\mathrm{Lip}_1(\mathbb R^k)$ if the fixed point set $\mathrm{Fix}(X,\mathbb R^k)$ embeds into $[0,1]$.
2. Any Borel action of a second countable locally compact group $G$ on a standard Borel space $X$ admits an injective $G$-equivariant Borel map into the shift space of $1$-Lipschitz functions from $G$ to the unit interval $\mathrm{Lip}_1(G)$.

Based on joint works with Qiang Huo and with Qiang Huo and Masaki Tsukamoto.