Recently a new class of continued fraction algorithms, the (N,α)-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each N∈N, N≥2 and α∈(0,N-1]. Each of these continued fraction algorithms has only finitely many possible digits. These (N,α)-expansions ‘behave’ very different from many other (classical) continued fraction algorithms; see also Chen and Kraaikamp (Matching of orbits of certain n-expansions with a finite set of digits (2022). To appear in Tohoku Math. J arXiv:2209.08882), de Jonge and Kraaikamp (Integers 23:17, 2023), de Jonge et al. (Monatsh Math 198(1):79–119, 2022), Nakada (Tokyo J Math 4(2):399–426, 1981) for examples and results. 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 α small enough, all rationals have an eventually periodic expansion with period 1. This happens for all α 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.

In [1], Boca and the fourth author of this paper introduced a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter α ∈ [g, G], where g = ¹₂ (^√5 − 1) and G = g + 1 = 1/g are the two golden mean numbers. Using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [1] are obtained, and it is shown that for each α, α^∗ ∈ [g, G] the natural extensions from [1] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for α ∈ [g, G], a fact already observed in [1]. Furthermore, it is shown that this approach can be extended to values of α smaller than g, and that for values of α ∈ [ ¹₆ (^√13 − 1), g] all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of α ∈ [0, G]. It is shown that if there exists an ergodic, absolutely continuous Tα-invariant measure, in any neighborhood of 0 we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. Moreover, we identify the largest interval on which the entropy is constant. In order to prove this we use a phenomenon called matching.

In this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss–Kuzmin–Lévy based approximation method is used. Convergence of this method follows from [32] but the speed of convergence remains unknown. For a lot of known densities the method gives a very good approximation in a low number of iterations. Finally, a subfamily of the N-expansions is studied. In particular, the entropy as a function of a parameter α is estimated for N=2 and N=36. Interesting behavior can be observed from numerical results.