Continued fractions and the d-dimensional Gauss transformation

Communications in Mathematical Physics

Volumn 215, Issue 3, 2001, Pages 487-515

  Hardcastle, D M ^a   Khanin, K ^a,b,c,d  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b ISAAC NEWTON INSTITUTE FOR MATHEMATICAL SCIENCES   (United Kingdom)

^c HEWLETT PACKARD LABORATORIES   (United Kingdom)

^d LANDAU INSTITUTE FOR THEORETICAL PHYSICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035530554     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200000290     Document Type: Article

References (27)

1. 1. Adler, R. and Flatto, L.: Cross section maps for geodesic flows. In: Ergodic Theory and Dynamical Systems, Progress in Mathematics 2, ed. A. Katok, Boston: Birkhäuser, 1980
2. 2. Bernstein, L.: The Jacobi-Perron Algorithm - Its Theory and Application. Lecture Notes in Mathematics 207, Berlin-Heidelberg-New York: Springer-Verlag, 1971
3. 3. Brun, V.: Algorithmes euclidiens pour trois et quatre nombres. In: 13 ième Congre. Math. Scand., Helsinki (1957), pp. 45-64
4. 4. Fujita, T., Ito, S., Keane, M. and Ohtsuki, M.: On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: A corrected proof. Ergod. Th. and Dyn. Sys. 16, 1345-1352 (1996)
5. 5. Gordin, M.I.: Exponentially rapid mixing. Dokl. Akad. Nauk SSSR 196, 1255-1258 (1971); English translation: Soviet Math. Dokl. 12, 331-335 (1970)
6. 5. Gordin, M.I.: Exponentially rapid mixing. Dokl. Akad. Nauk SSSR 196, 1255-1258 (1971); English translation: Soviet Math. Dokl. 12, 331-335 (1970)
7. 6. Hardcastle, D.M. and Khanin, K.: On almost everywhere strong convergence of multidimensional continued fraction algorithms. To appear in Ergod. Th. and Dyn. Sys.
8. 7. Ito, S., Keane, M. and Ohtsuki, M.: Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm. Ergod. Th. and Dyn. Sys. 13, 319-334 (1993)
9. 8. Ito, S. and Nakada, H.: On natural extensions of transformations related to Diophantine approximations. In: Number Theory and Combinatorics, Singapore: World Scientific, 1985, pp. 185-207
10. 9. Ito, S. and Yuri, M.: Number theoretical transformations with finite range structure and their ergodic properties. Tokyo J. Math. 10, 1-32 (1987)
11. 10. Jacobi, C.G.J.: Allgemeine Theorie der Kettenbruchähnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird. J. Reine Angew. Math. 69, 29-64 (1868)
12. 11. Khalatnikov, I.M., Lifshitz, E.M., Khanin, K.M., Shchur, L.N. and Sinai, Ya.G.: On the stochasticity in relativistic cosmology. J. Stat. Phys. 38, 97-114 (1985)
13. 12. Khinchin, A.Ya.: Continued Fractions. Chicago, Ill: University of Chicago Press, 1964
14. 13. Kontsevich, M.L. and Suhov, Yu.M.: Statistics of Klein polyhedra and multidimensional continued fractions. In: Pseudoperiodic Topology, eds. V. Arnold, M. L. Kontsevich and A. Zorich, Amer. Math. Soc. Transl. Series 2 197, 9-28 (1999)
15. 14. Lagarias, J.C.: The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms. Mh. Math. 115, 299-328 (1993)
16. k. Commun. Math. Phys. 95, 1-15 (1984)
17. 16. Nakada, H., Ito, S. and Tanaka, S.: On the invariant measure for the transformations associated with some real continued-fractions. Keio Eng. Rep. 30, 159-175 (1977)
18. 17. Perron, O.: Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64, 1-76 (1907)
19. 18. Podsypanin, E.V.: A generalization of the algorithm for continued fractions related to the algorithm of Viggo Brunn. Zap. Naucn. Sem. Leningrad Otdel. Mat. Inst. Steklov 67, 184-194 (1977); English translation: J. Soviet Math. 16, 885-893 (1981)
20. 18. Podsypanin, E.V.: A generalization of the algorithm for continued fractions related to the algorithm of Viggo Brunn. Zap. Naucn. Sem. Leningrad Otdel. Mat. Inst. Steklov 67, 184-194 (1977); English translation: J. Soviet Math. 16, 885-893 (1981)
21. 19. Poincaré, H.: Sur une généralisation des fractions continues. C. R. Acad. Sci. Paris 99, 1014-16 (1884)
22. 20. Rohlin, V.A.: Exact endomorphisms of a Lebesgue space. Amer. Math. Soc. Transl. Series 2 39, 1-36 (1964)
23. 21. Schweiger, F.: The Metrical Theory of the Jacobi-Perron Algorithm. Lecture Notes in Mathematics 334, Berlin-Heidelberg-New York: Springer-Verlag, 1973
24. 22. Schweiger, F.: A modified Jacobi-Perron algorithm with explicitly given invariant measure. In: Ergodic Theory, Proceedings Oberwolfach, Germany 1978, Lecture Notes in Mathematics 729, , Berlin-Heidelberg-New York: Springer-Verlag, 1979, pp. 199-202
25. 23. Selmer, E.: Om flerdimensjonal Kjede brøk. Nordisk Mat. Tidskr. 9, 37-43 (1961)
26. 24. Series, C.: On coding geodesics with continued fractions. Enseign. Math. 29, 67-76 (1980)
27. 25. Series, C.: The modular surface and continued fractions. J. London Math. Soc. (2) 31, 69-80 (1985)