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Multidimensional continued fractions and stable laws. (Fractions continues multidimensionnelles et lois stables.) (French) Zbl 0857.11035
The Jacobi-Perron algorithm is a multidimensional version of the classical continued fraction expansion. Starting from an $$x\in [0,1]^d$$ one computes $$\overline x = (x_2/x_1, x_3/x_1, \dots, 1/x_1)$$, and $$a(x) = [\overline x]$$, $$Tx = \{\overline x\}$$, where $$[ ]$$ and $$\{ \}$$ denote the integer and fractional part, respectively. The first approximant of $$x$$ is then defined as $$(1/a_d,a_1/a_d, \dots, a_{d-1}/a_d)$$, and the rest is $$Tx$$. One has $$x=(1/(a_d + (Tx)_d)$$, $$(a_1+(Tx)_1)/(a_d+(Tx)_d), \dots, (a_{d-1}+(Tx)_{d-1})/(a_d+(Tx)_d))$$. Continuing this procedure for all $$T^kx$$ as in the one dimensional case, one gets simultaneous rational approximations of the numbers $$x_1,\dots,x_d$$ of the form $$r_n=(p^1_n/q_n, \dots, p^d_n/q_n)$$. This algorithm relies on the iteration of the transformation $$T$$. It is well known (and goes back essentially to Gauss) that in the one dimensional case the measure $$T^n \lambda$$, where $$\lambda$$ is the Lebesgue measure on $$[0,1]$$, converges towards the measure $${1 \over \log 2} {dx \over 1+x}$$ which is the unique probability measure invariant under the transformation $$T$$, and absolutely continuous with respect to Lebesgue measure. This fact is extended in the paper to the multidimensional Jacobi-Perron algorithm, the density being now a piecewise analytic function on $$[0,1]$$.
Next the author considers the problem of the size of the integer vectors $$a(T^nx)$$, which are the coefficients in the continued fraction expansion of $$x$$, as $$n\to\infty$$. It is shown that for all multi-indices $$\alpha=(\alpha_1, \dots, \alpha_d)$$ where the $$\alpha_j$$ are positive real numbers, and $$\overline \alpha=\sum_j \alpha_j>1/2$$, the random variables $$1/n^{\overline \alpha} \sum^{n-1}_{k=0} (-1)^k a(T^k x)^\alpha$$ (where $$x$$ is chosen with Lebesgue measure) converge in distribution towards a multidimensional symmetric stable distribution of index $$1/ \overline \alpha$$. This extends a result due to Paul Lévy in the one-dimensional case.
Reviewer: Ph.Biane (Paris)

##### MSC:
 11J70 Continued fractions and generalizations 37A99 Ergodic theory 60F05 Central limit and other weak theorems
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##### References:
 [1] BERNSTEIN (L.) . - The Jacobi-Perron algorithm, its theory and application , Lecture Notes in Mathematics 207, Springer 1971 . MR 44 #2696 | Zbl 0213.05201 · Zbl 0213.05201 [2] BINGHAM (N.H.) , GOLDIE (C.M.) et TEUGELS (J.-L.) . - Regular variation . - Encyclopedia of Mathematics and its Applications, 1987 . MR 88i:26004 | Zbl 0617.26001 · Zbl 0617.26001 [3] BREIMAN (L.) . - Probability . - Addison-Wesley, 1968 . MR 37 #4841 | Zbl 0174.48801 · Zbl 0174.48801 [4] BIRKHOFF (G.) . - Extensions of Jentzsch’s theorem , Trans. Amer. Math. Soc., t. 104, 1965 , p. 507-512. MR 31 #4744 · Zbl 0209.19503 [5] BRENTJES (A.J.) . - Multi-dimensional continued fraction algorithms . - Mathematical centre tracts 145, Mathematisch Centrum, Amsterdam, 1981 . MR 83b:10038 | Zbl 0471.10024 · Zbl 0471.10024 [6] BUSEMANN (H.) and KELLY (P.) . - Projective geometry and projective metrics . - Academic Press, New York, 1953 . MR 14,1008e | Zbl 0052.37305 · Zbl 0052.37305 [7] DUNFORD (N.) et SCHWARTZ (J.T.) . - Linear operators, part I . - Interscience New York, 1958 . · Zbl 0080.00413 [8] FELLER (W.) . - An Introduction to probability theory and its applications Vol. II . - Willey, New York, 1966 . MR 35 #1048 | Zbl 0138.10207 · Zbl 0138.10207 [9] GUIVARC’H (Y.) et HARDY (J.) . - Théorèmes limites pour une classe de chaîne de Markov et applications aux difféomorphismes d’Anosov , Ann. Inst. H. Poincaré, t. 24, 1988 , p. 73-98. Numdam | MR 89m:60080 | Zbl 0649.60041 · Zbl 0649.60041 [10] GUIVARC’H (Y.) and LE JAN (Y.) . - Asymptotic winding of the geodesic flow on modular surfaces and continuous fractions , Ann. Sci. École Norm. Sup., t. 26, 1993 , p. 23-50. Numdam | MR 94a:58157 | Zbl 0784.60076 · Zbl 0784.60076 [11] KHINTCHINE (A. Ya.) . - Continued Fractions . - The University of Chicago Press, Chicago, 1964 . Zbl 0117.28601 · Zbl 0117.28601 [12] KRASNOLSELSKII (M.) . - Positive solutions of operator equations . - Groningen Noodshoff, 1964 . · Zbl 0121.10604 [13] KREIN (M.G.) and RUTMAN (M.A.) . - Linear operators leaving invariant a cone in a Banach space , Translations A.M.S., series one, vol. 10, Functional Analysis and Measure Theory, p. 199-325, 1962 . [14] LÉVY (P.) . - Théorie de l’addition des variables aléatoires . - Gauthier-Villars, Paris, 1937 . Zbl 0016.17003 | JFM 63.0490.04 · Zbl 0016.17003 [15] LÉVY (P.) . - Sur le développement en fraction continue d’un nombre choisi au hasard , Compositio Math., t. 3, 1936 , p. 286-303. Numdam | Zbl 0014.26803 | JFM 62.0246.01 · Zbl 0014.26803 [16] MAYER (D.) . - Approach to equilibrium for locally expanding maps in Rk , Comm. Math. Phys., t. 95, 1984 , p. 1-15. Article | MR 86d:58069 | Zbl 0577.58022 · Zbl 0577.58022 [17] MONTEL (P.) . - Leçons sur les familles normales de fonctions analytiques et leur application , chap. I et IX. - Gauthier-Villar Paris, 1927 . JFM 53.0303.02 · JFM 53.0303.02 [18] ROUSSEAU-EGÈLE (J.) . - Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux , Ann. Probab., t. 3, 1983 , p. 772-788. Article | MR 84m:60032 | Zbl 0518.60033 · Zbl 0518.60033 [19] SCHWEIGER (F.) . - The metrical theory of Jacobi-Perron algorithm , Lecture Notes in Mathematics 334, Springer 1973 . MR 49 #10654 | Zbl 0287.10041 · Zbl 0287.10041 [20] ZYGMUND (A.) . - Trigonometrical series , vol. I. - Second edition, Cambridge University Press, 1959 . · JFM 61.0263.03
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