an:01902589 Zbl 1022.11034 Hardcastle, D. M. The three-dimensional Gauss algorithm is strongly convergent almost everywhere EN Exp. Math. 11, No. 1, 131-141 (2002). 00083056 2002
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11J70 11K50 multidimensional continued fractions; Brun's algorithm; Jacobi-Perron algorithm; strong convergence; Lyapunov exponents Let $$X=\{(x_1,x_2,x_3)\in [0,1]^3\mid x_1\geq x_2\geq x_3\}$$ and define the three-dimensional ordered Jacobi-Perron algorithm(JPA) $$T:X\to X$$ as $T(x_1,x_2,x_3)= \begin{cases} (\{\frac 1{x_1}\},\frac{x_2}{x_1},\frac{x_3}{x_1}) & \text{if } \{\frac 1{x_1}\}>\frac{x_2}{x_1},\\ (\frac{x_2}{x_1},\{\frac 1{x_1}\},\frac{x_3}{x_1}) & \text{if } \frac{x_2}{x_1}>\{\frac 1{x_1}\}>\frac{x_3}{x_1},\\ (\frac{x_2}{x_1},\frac{x_3}{x_1},\{\frac 1{x_1}\}) & \text{if } \frac{x_3}{x_1}>\{\frac 1{x_1}\}, \end{cases}$ where $$\{x\}$$ denotes the fractional part of $$x$$. This is also called the Gauss algorithm, and it is equivalent to Brun's algorithm and to the modified JPA. It is proved that the three-dimensional Gauss algorithm is strongly convergent almost everywhere on $$X$$. The proof involves the computer assisted estimation of the largest Lyapunov exponent of a cocycle associated to the algorithm. M.Ohtsuki (Kodaira)