Idempotents in the Ellis semigroup of Floyd-Auslander systems

We study minimal idempotents $J^{\mathrm{min}}(X)$ in the Ellis semigroup $E(X)$ associated with a Floyd-Auslander system $(X,T)$. We show that $(X,T)$ is non-tame if and only if $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$, which happens exactly when the factor map onto the maximal equicontinuous factor possesses uncountably many non-invertible fibres. This yields an easy-to-check criterion for distinguishing tame from non-tame Floyd-Auslander systems and, more importantly, provides an entire family of regular almost automorphic systems with $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$. Notably, all previously known regular almost automorphic non-tame systems exhibited only a small (i.e. $\leq 2^{\aleph_0}$) set of minimal idempotents. We obtain our result by leveraging an alternative characterisation of (non)-tameness through, what we call, choice domains.