Calta–Kraaikamp–Schmidt natural-extension conjecture for triangle-group continued fractions

For each integer $n\geq 3$ and parameter $\alpha\in(0,1)$, let $T_{n,\alpha}$ be the continued fraction map associated with the triangle Fuchsian group $G_n$, and let its first pointwise expansive power be the first iterate that is pointwise expansive. A natural extension is an invertible measure-preserving system projecting onto the original system.

Calta–Kraaikamp–Schmidt conjecture. For all $n\geq 3$ and all $\alpha\in(0,1)$, the natural extension of the first pointwise expansive power of $T_{n,\alpha}$ is given by the first return of the geodesic flow to a cross section in the unit tangent bundle of the hyperbolic orbifold uniformized by $G_n$.

The conjecture connects these continued fraction transformations with geodesic-flow codings. The paper proves the relevant entropy and natural-extension results and establishes an equivalent volume identity, but the asserted identification is presented as conjectural.