International audienceFrom the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base~$\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits

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

We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the Rényi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces as well as the dynamical properties of the system.2010 Mathematics Subject Classification. Primary: 37C40, 11K50.

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