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The three-dimensional Gauss algorithm is strongly convergent almost everywhere. (English) Zbl 1022.11034
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.

MSC:
 11J70 Continued fractions and generalizations 11K50 Metric theory of continued fractions
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References:
 [1] Baldwin P. R., Jour. Stat. Phys. 66 pp 1463– (1992) · Zbl 0891.11039 · doi:10.1007/BF01054430 [2] Baldwin P. R., Jour. Stat. Phys. 66 pp 1507– (1992) · Zbl 0890.11024 · doi:10.1007/BF01054431 [3] Brun V., 13 ième Congre. Math. Scand., Helsinki pp 45– (1957) [4] Fujita T., Ergod. Th. and Dyn. Sys. 16 pp 1345– (1996) · Zbl 0868.28008 · doi:10.1017/S0143385700010063 [5] Hardcastle D. M., Ergod. Th. and Dyn. Sys. 20 pp 1711– (2000) · Zbl 0977.11031 · doi:10.1017/S014338570000095X [6] Hardcastle D. M., Commun. Math. Phys. 215 pp 487– (2001) · Zbl 0984.11040 · doi:10.1007/s002200000290 [7] Hardcastle D. M., Experimental Mathematics 11 (1) pp 119– (2002) · Zbl 1029.11037 · doi:10.1080/10586458.2002.10504474 [8] Ito S., Ergod. Th. and Dyn. Sys. 13 pp 319– (1993) [9] Jacobi C. G. J., J. Reine Angew. Math. 69 pp 29– (1868) · JFM 01.0062.01 · doi:10.1515/crll.1868.69.29 [10] Khanin K., Talk at the International Workshop on Dynamical Systems (1992) [11] Kosygin D. V., Advances in Soviet Mathematics 3 pp 99– (1991) [12] Kingman J. F. C., J. Royal Stat. Soc. B 30 pp 499– (1968) [13] Lagarias J. C., Mh. Math. 115 pp 299– (1993) · Zbl 0790.11059 · doi:10.1007/BF01667310 [14] Meester R., Ergod. Th. and Dyn. Sys. 19 pp 1077– (1999) · Zbl 1044.11074 · doi:10.1017/S0143385799133960 [15] Oseledets V. I., Trans. Moscow Math. Soc. 19 pp 197– (1968) [16] Paley R. E. A. C., Proc. Cambridge Philos. Soc. 26 pp 127– (1930) · JFM 56.1053.06 · doi:10.1017/S0305004100015371 [17] Perron O., Math. Ann. 64 pp 1– (1907) · JFM 38.0262.01 · doi:10.1007/BF01449880 [18] Podsypanin E. V., Zap. Naucn. Sem. Leningrad Otdel. Mat. Inst. Steklov 67 pp 184– (1977) [19] Press W. H., Numerical Recipes in C: The Art of Scientific Computing, Second Edition (1992) · Zbl 0845.65001 [20] Schratzberger B. R., Sitzungsber. AM. II 207 pp 229– (1998) [21] Schweiger F., Ergodic Theory, Proceedings Oberwolfach, Germany 1978, Lecture Notes in Mathematics 729 pp 199– (1979) [22] Schweiger, F. ”The exponent of convergence for the 2-dimensional Jacobi-Perron algorithm,”. Proceedings of the Conference on Analytic and Elementary Number Theory in Vienna 1996. Edited by: Nowak, W. G. and Schoissengeier, J. pp.207–213. [Schweiger 96] · Zbl 0879.11044 [23] Selmer E., Nordisk Mat. Tidskr. 9 pp 37– (1961)
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