Arithmetic-dynamic simplicity dichotomy for geodesics

Consider a lattice $F\subset \operatorname{PSL}_2(\mathbb{R})$ with invariant trace field $\K=\mathbb{Q}(\{\operatorname{disc}(\gamma)\mid \gamma\in F\})$. Let $S=F\backslash\mathbb{H}$, with mapping class group $\operatorname{Mod}(S)=\operatorname{Out}(F)$. Fix a simple geodesic $\alpha\subset S$, let $\widetilde{\alpha}\subset\mathbb{H}$ be its lift, viewed as an $F$-invariant set of geodesics, and let $\partial\widetilde{\alpha}\subset\mathbb{R}\mathbb{P}^1$ be the set of their endpoints. Arithmetic-dynamic simplicity conjecture. The following dichotomy holds: if $\alpha$ is fixed by an element of infinite order in $\operatorname{Mod}(S)$, then the endpoints in $\partial\widetilde{\alpha}$ are algebraic over $\K$; if $\alpha$ is not fixed by any element of infinite order in $\operatorname{Mod}(S)$, then the endpoints in $\partial\widetilde{\alpha}$ are transcendental over $\K$. In the first case, the relevant algebraic extension is further determined by the type of infinite-order mapping class: for a maximal product of commuting Dehn twists, the lifted geodesics are bi-asymptotic to simple closed geodesics and their endpoints lie in a quadratic extension of $\K$; for a pseudo-Anosov element $A\in\operatorname{Mod}(S)$, the endpoints are expected to belong to $\K[\mu_j]$ for a Lyapunov exponent $\mu_j$ given by a root of the Teichmüller polynomial of $A$. This proposes an arithmetic/transcendence dichotomy for simple geodesics, relating the algebraicity of their endpoints to dynamical symmetries of the surface. The general status is not supplied in the source.