We study multifractal spectra of a simple class of interval transformations with a neutral fixed point. The analyticity of their spectral curves depends on whether the absolutely continuous invariant measure is infinite or not.

1. Statements of results. By the paper of Lagarias [4], the metrical convergence of algorithms of simultaneous Diophantine approximation has been clearly formulated as a problem of ergodic theory and dynamical systems. Since then, much of the study was devoted to variants of the Jacobi-Perron algorithm, which is a direct generalisation of the continued fraction algorithm. There is another class of algorithms which includes the algorithms of Selmer and Brun. This class appears to be more practical since the operations of approximation consist of addition (subtraction) instead of multiplication as in the case of the Jacobi-Perron type algorithms. In this paper, we prove the almost everywhere strong convergence of a class of two-dimensional algorithms of additive type (Corollary 2). The problem of strong convergence concerns the speed of convergence of Diophantine quantities q n x − p n , which typically attenuate oscillating as n goes to infinity. For future reference, we formulate our criterion for arbitrary finite-dimensional algorithms (Theorem 1).Metric properties of Selmer's algorithm have already been studied by Schweiger [5]. Schweiger reduces Selmer's algorithm by the jump transformation to Baldwin's one, the ergodic properties of which are already established. However, his argument covers one particular sequence of fractions of approximation. Our method covers all the fractions given by the algorithm, which is complementary to Schweiger [5] in this respect, and as a consequence, we obtain information on the second Lyapunov exponent of the system. For a proof of the strong convergence of two-dimensional Brun's algorithm for the multiplicative case (cf. Remark 1

Convergence properties of multidimensional continued fraction algorithms introduced by V. Baladi and A. Nogueira are studied. The paper contains an arithmetic proof of almost everywhere exponentially strong convergence of some two-dimensional Markovian random algorithms and dynamically defined ones. A special three-dimensional deterministic case is also discussed.

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