A length comparison theorem for geodesic currents

We work with the space $\mathcal C(S)$ of geodesic currents on a closed surface $S$ of negative Euler characteristic. By prior work of the author with Sebastian Hensel, each filling geodesic current $\mu$ has a unique length-minimizing metric $X$ in Teichm\"uller space. In this paper, we show that, on so-called thick components of $X$, the geometries of $\mu$ and $X$ are comparable, up to a scalar depending only on $\mu$ and the topology of $S$. We also characterize thick components of the projection using only the length function of $\mu$.