For β ∈ (1, 2] the β-transformation T β : [0, 1) → [0, 1) is defined by T β (x) = βx (mod 1). For t ∈ [0, 1) let K β (t) be the survivor set of T β with hole (0, t) given byfor all n ≥ 0 . In this paper we characterise the bifurcation set E β of all parameters t ∈ [0, 1) for which the set valued function t → K β (t) is not locally constant. We show that E β is a Lebesgue null set of full Hausdorff dimension for all β ∈ (1, 2). We prove that for Lebesgue almost every β ∈ (1, 2) the bifurcation set E β contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of β ∈ (1, 2) for which E β contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for E 2 , the bifurcation set of the doubling map. Finally, we give for each β ∈ (1, 2) a lower and upper bound for the value τ β , such that the Hausdorff dimension of K β (t) is positive if and only if t < τ β . We show that τ β ≤ 1 − 1 β for all β ∈ (1, 2).Urbański proved that the function t → h top (g|K g (t)) is a Devil's staircase, where h top denotes the topological entropy.Motivated by the work of Urbański, we consider this situation for the β-transformation. Given β ∈ (1, 2], the β-transformation T β : [0, 1) → [0, 1) is defined by T β (x) = βx

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 [1] 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.

Two closely related families of α-continued fractions were introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other hand. The behavior of the entropy as a function of the parameter α has been studied extensively for Nakada’s family, and several of the results have been obtained exploiting an algebraic feature called matching. In this article we show that matching occurs also for Tanaka–Ito α-continued fractions, and that the parameter space is almost completely covered by matching intervals. Indeed, the set of parameters for which the matching condition does not hold, called the bifurcation set, is a zero measure set (even if it has full Hausdorff dimension). This property is also shared by Nakada’s α-continued fractions, and yet there also are some substantial differences: not only does the bifurcation set for Tanaka–Ito continued fractions contain infinitely many rational values, it also contains numbers with unbounded partial quotients.

A new continued fraction expansion algorithm, the so-called a/b-expansion, is introduced and some of its basic properties, such as convergence of the algorithm and ergodicity of the underlying dynamical system, have been obtained. Although seemingly a minor variation of the regular continued fraction (RCF) expansion and its many variants (such as Nakada's α-expansions, Schweiger's odd-and even-continued fraction expansions, and the Rosen fractions), these a/b-expansions behave very differently from the RCF and many important question remains open, such as the exact form of the invariant measure, and the "shape" of the natural extension.

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As a natural counterpart to Nakada's α-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite σ-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.

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 = 1 2 ( √ 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 ∈ [ 1 6 ( √ 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 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. In order to prove this we use a phenomena called matching.Recently, Boca and the fourth author introduced in [1] a new class of continued fraction expansions: α-continued fraction expansions with odd partial quotients. These α-expansions are reminiscent of Nakada's α-expansions from [14], and are studied for a certain range of the parameter (in case of Nakada: α ∈ [ 1 2 , 1], in case

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