(non)-matching and (non)-periodicity for $(N,\alpha)$-expansions

Recently a new class of continued fraction algorithms, the $(N,\alpha$)-expansions, was introduced for each $N\in\mathbb{N}$, $N\geq 2$ and $\alpha \in (0,\sqrt{N}-1]$. Each of these continued fraction algorithms has only finitely many possible digits. These $(N,\alpha)$-expansions `behave' very different from many other (classical) continued fraction algorithms. In this paper we will show that when all digits in the digit set are co-prime with $N$, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for $\alpha$ small enough, all rationals have an eventually periodic expansion with period 1. This happens for all $\alpha$ when $N=2$. We also find infinitely many matching intervals for $N=2$, as well as rationals that are not contained in any matching interval.