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Zbl 1012.11060
Broise-Alamichel, Anne; Guivarc'h, Yves
Exposants caractéristiques de l'algorithme de Jacobi-Perron et de la transformation associée. (Characteristic exponents of the Jacobi-Perron algorithm and of the associated map). (French)
[J] Ann. Inst. Fourier 51, No.3, 565-686 (2001). ISSN 0373-0956; ISSN 1777-5310/e

This paper proves the simplicity of the spectrum of the Lyapunov exponents for some classical multidimensional continued fraction algorithms such as the Jacobi-Perron algorithm and the Brun algorithm. The main part of the paper is devoted to the Jacobi-Perron case, and an appendix gives the corresponding proofs for the Brun algorithm. \par The authors first recall some classical properties of the Jacobi-Perron algorithm, such as the existence of an analytic invariant probability measure absolutely continuous with respect to the Lebesgue measure [see also {\it A. Broise}, Bull. Soc. Math. Fr. 124, 97-139 (1996; Zbl 0857.11035)]. A geometric description is detailed inspired by Poincaré's algorithm for the usual continued fractions; special focus is given on the projective version of the algorithm in terms of products of matrices and on the simple geometric construction of the successive approximations of a point by this algorithm.\par Indeed the $d$-dimensional Jacobi-Perron algorithm associates with $x\in [0,1]^d$ a sequence of matrices with nonnegative coefficients of size $d+1$ which converge towards a matrix of the image space generated by the direction $(x,1)$. The product of the first $n$ matrices generates the coordinates of a basis $(e_{n-d},\dots,e_n)$ of the lattice $\bbfZ^{d+1}$ such that their projection on a suitable hyperplane produces a sequence of nested simplices $\sigma_n(x) = (p_n/q_n,\dots,p_{n+d}/q_{n+d})$ which contain and converge to $x$, their edges being rational approximations of $x$.\par Then the Lyapunov exponents (satisfying $\lambda_1\geq \lambda_2\geq \dots \geq\lambda_{d+1}$ and $\lambda_1+\cdots+\lambda_{d+1}= 0$) are introduced for the Jacobi-Perron algorithm, expressing the almost everywhere asymptotic form of the simplices $\sigma_n(x)$.\par Special attention is devoted to the case of dimension 2 and to the proof of $\lambda_2 < 0$, one of the main results of the paper. The application to the simultaneous approximation of two reals by the Jacobi-Perron algorithm is also discussed, by deducing the a.e. exponential convergence of the algorithm. In the Brun's case, one recovers the result of {\it T. Fujita, S. Ito, M. Keane} and {\it M. Ohtsuki}, Ergodic Theory Dyn. Syst. 16, 1345-1352 (1996; Zbl 0868.28008)].\par The rest of the paper is then devoted to the higher-dimensional case, and to the proof of the main result, that is, $\lambda_1 >\lambda_2 >\cdots >\lambda_{d+1}$ and $\lambda_1 + \lambda_{d+1} > 0$. In geometric terms, this implies that the volumes of the $(d-1)$-faces of the simplices $\sigma_n(x)$ converge to 0 exponentially fast. Nothing can be said about the distance between one point $x$ and its $n$th-approximation, which illustrates the fact that the Jacobi-Perron algorithm does not provide optimal simultaneous approximations.\par The results are obtained by first stating a generalization of the inequality by {\it R. E. A. C. Paley} and {\it H. D. Ursell} [Proc. Camb. Philos. Soc. 26, 127-144 (1930; JFM 56.1053.06)], from which one deduces the decrease of the sizes of the triangles (in dimension 2) or of the volumes of the faces of the simplices (in the general case). A careful study of some cohomological equations is then conducted by introducing transfer operators and using sharp results of ergodic theory, as well as a proof for the fact that the Zariski closure of the semigroup generated by the Jacobi-Perron matrices contains $\text{SL}(d +1, \bbfZ)$.\par This rich paper (more than one hundred pages) is not only remarkable for its results (there exist only few examples of simplicity results for the Lyapunov exponents) but also an excellent introduction to the Jacobi-Perron algorithm thanks to the clear writing of the proofs and to the evocation of the underlying geometric ideas which provides an intuitive approach of the proofs.
[Valérie Berthé (Marseille)]

MSC 2000:: *11J70 Continued fractions and generalizations
11K50 Metric theory of continued fractions
37H15 Multiplicative ergodic theory, Lyapunov exponents
11J13 Simultaneous homogeneous approximation, linear forms

Keywords: transfer operators; product of random stationary matrices; Lyapunov spectrum; Jacobi-Perron algorithm; periodic points; simplicity of the spectrum; multidimensional continued fraction algorithms; Brun algorithm; simultaneous approximation of two reals; almost everywhere exponential convergence

Citations: Zbl 0857.11035; Zbl 0868.28008; JFM 56.1053.06

Cited in: Zbl 1172.11022

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