Minimality conjecture for the remaining affine interval exchange transformations

Let $F_t$ be the affine interval exchange transformation associated with the parameter $t$, and let $\Lambda_{\Gamma}\subset S^1$ be the Cantor set of remaining parameters, defined as the limit set of a Zariski-dense subgroup $\Gamma<\mathrm{SL}(2,\mathbb{R})$ of infinite covolume. Minimality conjecture. For every parameter $t\in\Lambda_{\Gamma}$, $F_t$ is minimal. The paper reports only partial results for parameters in $\Lambda_{\Gamma}$; the conjecture is motivated by numerical experiments and remains open.