Let $S= S_{g,n}$ be a finite type orientable surface of genus $g$ and $n$ punctures and let $mathcal{ML}(S)$ denote the corresponding space of measured laminations. The Thurston measure, $mu^{Th},$ is a mapping class group invariant and locally finite Borel measure on $mathcal{ML}(S)$ which is obtained as a weak-$star$ limit of (appropriately weighted and rescaled) sums of Dirac measures supported on the set of integral multi-curves.

The Thurston measure arises in Mirzakhani’s curve counting framework. Concretely, given a hyperbolic metric $rho$ on $S_{g,n}$, let $B_{rho} subset mathcal{ML}(S)$ denote the set of measured laminations with $rho$-length at most $1$. Then $mu^{Th}(B_{rho})$ controls the top coefficient of the polynomial that counts multi-curves up to a certain $rho$-length and living in a given mapping class group orbit.

**Question:** Fix a hyperbolic metric $rho$ on $S$ and a finite (not necessarily regular) cover $p: Y rightarrow S$. Let $rho_{p}$ denote the hyperbolic metric on $Y$ obtained by pulling $rho$ back to $Y$ via $p$. Is there a straightforward relationship between $mu^{Th}(B_{rho_{p}})$ and $mu^{Th}(B_{rho})$? For example, is the ratio

$$ frac{mu^{Th}(B_{rho})}{mu^{Th}(B_{rho_{p}})} $$

uniformly bounded away from $0$ and $infty$? Does it equal a fixed value, independent of $rho$? If so, can it be easily related to the degree of the cover $Y rightarrow S$?

It seems hard to approach the above by thinking about counting curves on $Y$ versus $S$, because “most” simple closed curves on $Y$ project to non-simple curves on $S$. But, maybe the generalizations of curve counting for non-simple curves due to Mirzakhani (https://arxiv.org/pdf/1601.03342.pdf) or Erlandsson-Souto (https://arxiv.org/pdf/1904.05091.pdf) could be useful. Of course, both apply to counting curves in a fixed mapping class group orbit, so it’s not clear (to me) how to apply these results either since multi-curves on $Y$ can project to curves on $S$ with arbitrarily large self intersection.

Thanks for reading! I appreciate any ideas or reading suggestions.