Exposants caractéristiques de l'algorithme de Jacobi-Perron et de la transformation associée

Broise-Alamichel, Anne; Guivarc’h, Yves

doi:10.5802/aif.1832

Cited by 27 publications

(23 citation statements)

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“…For more details, see for instance [13]. The dynamical system defined by ~ on the unit square has been much studied [40], in particular its invariant measures and its unique invariant ergodic probability measure equivalent to Lebesgue measure [14], [15] (generalization of the classical Gauss measure) .…”

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confidence: 99%

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Arnoux¹,

Berthé²,

Ito³

2002

Annales De L’institut Fourier

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confidence: 99%

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Arnoux¹,

Berthé²,

Ito³

2002

Annales De L’institut Fourier

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“…For a proof of the strong convergence of two-dimensional Brun's algorithm for the multiplicative case (cf. Remark 1.1), we refer to BroiseAlamichel and Guivarc'h [1]. For future use, we axiomatize our criterion for algorithms we consider to be strongly convergent.…”

Section: Statements Of Resultsmentioning

confidence: 99%

Strong convergence of additive Multidimensional Continued Fraction algorithms

Nakaishi

2006

Acta Arith.

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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

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“…However, this measure is not known explicitly for Jacobi-Perron (the density of the measure is shown to be a piecewise analytical function in [21]). For a thorough study of the Lyapounov exponents of the Jacobi-Perron algorithm, see [22]. Nevertheless, as underlined in [20], concerning the class of so-called vectorial algorithms to which Jacobi-Perron algorithm belongs, "All continued fraction algorithms which have been proposed since the beginning (Jacobi, 1868), and up to about 1970 belong to this class.…”

Section: Toward Multidimensional Expansionsmentioning

confidence: 99%

Numeration and discrete dynamical systems

Berthé

2011

Computing

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This survey aims at giving both a dynamical and computer arithmetic-oriented presentation of several classical numeration systems, by focusing on the discrete dynamical systems that underly them: this provides simple algorithmic generation processes, information on the statistics of digits, on the mean behavior, and also on periodic expansions (whose study is motivated, among other things, by finite machine simulations). We consider numeration systems in a broad sense, that is, representation systems of numbers that also include continued fraction expansions. These numeration systems might be positional or not, provide unique expansions or be redundant. Special attention will be payed to β-numeration (one expands a positive real number with respect to the base β > 1), to continued fractions, and to their Lyapounov exponents. In particular, we will compare both representation systems with respect to the number of significant digits required to go from one type of expansion to the other one, through the discussion of extensions of Lochs' theorem.

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Exposants caractéristiques de l'algorithme de Jacobi-Perron et de la transformation associée

Cited by 27 publications

References 1 publication

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Strong convergence of additive Multidimensional Continued Fraction algorithms

Numeration and discrete dynamical systems

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