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.@en

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property “bounded non-full range” used to relax traditional Markov conditions. Second, the “quilting” operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada α-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.@en

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.

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For N∈ N_{≥ 2} and α∈ R such that 0<α≤N-1, we define I_α: = [α, α+ 1] and Iα-:=[α,α+1) and investigate the continued fraction map Tα:Iα→Iα-, which is defined as Tα(x):=Nx-d(x), where d: I_α→ N is defined by d(x):=⌊Nx-α⌋. For N∈ N_{≥ 7}, for certain values of α, open intervals (a, b) ⊂ I_α exist such that for almost every x∈ I_α there is an n∈ N for which Tαn(x)∉(a,b) for all n≥ n. These gaps (a, b) are investigated using the square Υα:=Iα×Iα-, where the orbitsTαk(x),k=0,1,2,… of numbers x∈ I_α are represented as cobwebs. The squares Υ_α are the union of fundamental regions, which are related to the cylinder sets of the map T_α, according to the finitely many values of d in T_α. In this paper some clear conditions are found under which I_α is gapless. If I_α consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of I_α with regard to the fixed points of I_α under T_α.

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In 1991 a new class of continued fraction expansions, the S-expansions, was introduced in this journal. This class contains many classical continued fraction algorithms, such as Nakada’s α-expansions (for α between 1/2 and 1), the nearest integer continued fraction, Minkowski’s diagonal continued fraction expansion, and Bosma’s optimal continued fraction. These S-expansions were obtained from the natural extension of the regular continued fraction (RFC) via induced transformations. Therefore many metric and arithmetic properties of these S-expansions can be derived from the corresponding classical results on the RFC. In particular, the natural extensions of these S-expansions were obtained. The second coordinate map of these natural extensions is the inverse of a continued fraction algorithm. In this paper we study these ‘reversed algorithms’; in particular we show they are again S-expansions, and we find the corresponding singularization areas.@en

By means of singularisations and insertions in Nakada's α-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map Tα is given for (10-2)/3≤α<1. From our construction it follows that Ωα, the domain of the natural extension of Tα, is metrically isomorphic to Ωg for α∈[g2,g), where g is the small golden mean. Finally, although Ωα proves to be very intricate and unmanageable for α∈[g2,(10-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.

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Denote by p n /q n ,n=1,2,3,…, pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of a real irrational number x x . Define the sequence of approximation coefficients by θ n (x):=q n |q n x−p n |,n=1,2,3,… θn(x):=qn|qnx−pn|,n=1,2,3,… . In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as θ n−1 <θ n <θ n+1  θn−1<θn<θn+1 , occur for almost all x x with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.@en

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.

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We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.@en