Entropy-volume conjecture for triangle-group continued fraction transformations

Fix an integer $n\geq 3$ and a parameter $\alpha\in(0,1)$. Let $T_\alpha$ be the corresponding continued fraction transformation, let $\mu$ be the invariant measure on its natural-extension domain $\Omega_\alpha$, and let $\operatorname{vol}_n$ denote the volume of the unit tangent bundle of the hyperbolic orbifold uniformized by the triangle group $G_n$:

$$\operatorname{vol}_n=\frac{2(2n-3)\pi^2}{3n}.$$

Entropy-volume conjecture. For all $n\geq 3$ and all $\alpha\in(0,1)$,

$$h(T_\alpha)\,\mu(\Omega_\alpha)=\operatorname{vol}_n.$$

The identity is equivalent in the paper to the conjectured geometric realization of the first pointwise expansive power as a geodesic-flow return map. It is proved on certain parameter intervals and numerically confirmed for $4\leq n\leq 12$, but remains conjectural in full generality.