Characteristic exponents of the Jacobi-Perron algorithm and of the associated map

 Access to full text

 Full (PDF)

1. P. Arnoux, A. Nogueira, Mesures de Gauss pour des algorithmes de fractions continues, Ann. École Norm. Sup. 26 (1993), 645-664 Zbl0801.11036MR1251147
2. G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc. 13 (1976), 486-488 Zbl0342.60049MR419727
3. P.R. Baldwin, A multidimensional continued fractions and some of its statistical properties, J. Stat. Physics 66 (1992), 1463-1505 Zbl0891.11039MR1156411
4. P.R. Baldwin, A convergence exponent for multidimensional continued fractions algorithms, J. Stat. Physics 66 (1992), 1507-1526 Zbl0890.11024MR1156412
5. A. Borel, Introduction aux groupes arithmétiques, (1969), Hermann, Paris Zbl0186.33202MR244260
6. P. Bougerol, J. Lacroix, Products of random matrices with applications to Schrödinger operators, 8 (1985), Birkhäuser Zbl0572.60001MR886674
7. A. Broise, (1994) 
8. A. Broise, Fractions continues multidimensionnelles et lois stables, Bull. Soc. Math. France 124 (1996), 97-139 Zbl0857.11035MR1395008
9. J.W.S. Cassels, An introduction to diophantine approximation, (1957), Cambridge University Press, Cambridge Zbl0077.04801MR87708
10. J.-P. Conze, Y. Guivarc'h, Limits sets of groups of linear transformations, Ergodic Theory and Harmonic Analysis 62, Pt 3 (2000), 367-385 Zbl1115.37305
11. J.-P. Conze, A. Raugi, Fonctions harmoniques pour un opérateur de transition et applications, Bull. Soc. Math. France 118 (1990), 273-310 Zbl0725.60026MR1078079
12. S.G. Dani, Dynamical systems on homogeneous spaces, Math. Physics I: Dynamical systems, Ergodic theory and Applications vol. 100, chap. 6 (2000), Springer Zbl06389526
13. H. Furstenberg, Non commuting random products, Trans. Amer. Math. Soc. 108 (1963), 377-428 Zbl0203.19102MR163345
14. I. Goldsheid, Y. Guivarc'h, Zariski closure and the dimension of the Gaussian law of the product of random matrices. I, Probab. Theory Relat. Fields 105 (1996), 109-142 Zbl0854.60006MR1389734
15. I. Goldsheid, G.A. Margulis, Simplicity of the Liapunoff spectrum for product of random matrices, Soviet Math. 35 (1987), 309-313 Zbl0638.15010
16. M. Gordin, Exponentially fast mixing, Sov. Math. Dokl. 12 (1971), 331-335 Zbl0269.60028
17. W.L. Greenberg, Discrete groups with dense orbits, Flows on homogeneous spaces (1963), 85-103, Princeton University Press, Princeton 
18. Y. Guivarc'h, Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés, Ergodic Th. Dynam. Systems 9 (1989), 433-453 Zbl0693.58011MR1016662
19. Y. Guivarc'h, Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Th. Dynam. Systems 10 (1990), 483-512 Zbl0715.60008MR1074315
20. Y. Guivarc'h, E. Le Page, Transformée de Laplace d'une mesure de probabilité sur le groupe linéaire et applications, (2000) 
21. Y. Guivarc'h, A. Raugi, Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahr. Verw. Geb. 69 (1985), 187-242 Zbl0558.60009MR779457
22. Y. Guivarc'h, A. Raugi, Product of random matrices: convergence theorems, Contemp. Math. 50 (1986), 31-54 Zbl0592.60015MR841080
23. Y. Guivarc'h, A. Raugi, Propriétés de contraction d’un semi-groupe de matrices inversibles. Cde Liapunoff d’un produit de matrices aléatoires indépendantes, Israel J. Math. 65 (1989), 165-196 Zbl0677.60007MR998669
24. D.M. Hardcastle, K. Khanin, On almost everywhere strong convergence of multidimensionnal continued fractions algorithms, Ergodic Th. Dynam. Systems 20 (2000), 1711-1733 Zbl0977.11031MR1804954
25. D.M. Hardcastle, K. Khanin, Continued fractions and the d-dimensionnal Gauss transformation, (2000) Zbl0984.11040MR1810942
26. C.T. Ionescu-Tulcea, G. Marinescu, Théorie ergodique pour des classes d'opérations non complètement continues, Ann. Math. 52 (1950), 140-147 Zbl0040.06502MR37469
27. S. Ito, M. Keane, M. Ohtsuki, Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm, Ergodic Th. Dynam. Systems 13 (1993), 319-334 Zbl0846.28005MR1235475
28. D.V. Kosygin, Multidimensional KAM theory from the renormalisation group viewpoint, Dynamical System and Statistical Mechanics 3 (1991), 99-130, AMS, Providence Zbl0731.58062
29. J.C. Lagarias, The quality of the diophantine approximations found by the Jacobi-Perron and related algorithms, Mh. Math. 115 (1993), 299-328 Zbl0790.11059MR1230366
30. J.C. Lagarias, Geodesic multidimensionnal continued fractions, Proc. Lond. Math. Soc., III 69 (1994), 464-488 Zbl0813.11040MR1289860
31. E. Le Page, Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications, Ann. Inst. Henri Poincaré 25 (1989), 109-142 Zbl0679.60010MR1001021
32. D. Mayer, Approach to equilibrium for locally expanding maps in ${ℝ}^k$, Comm. Math. Phys. 9 (1984), 1-15 Zbl0577.58022MR757051
33. R. Meester, A simple proof of the exponential convergence of the modified Jacobi-Perron algorithm, Ergodic Th. Dynam. Systems 19 (1999), 1077-1083 Zbl1044.11074MR1709431
34. P. Montel, Leçons sur les familles normales de fonctions analytiques et leurs applications, chap. I et IX (1927), Gauthier-Villars, Paris 
35. A. Nogueira, The three-dimensional Poincaré continued fractions algorithm, Israel J. Math. 90 (1995), 373-401 Zbl0840.11030MR1336331
36. V.I. Oseledets, A multiplicative ergodic theorem, Trans. Moscow Math. Soc. 19 (1968), 197-231 Zbl0236.93034MR240280
37. R.E.A.C. Paley, H.D. Ursell, Continued fractions in several dimensions, Proc. Camb. Phil. Soc. 26 (1930), 127-144 Zbl56.1053.06
38. O. Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus, Math. Ann. 64 (1907), 1-76 Zbl38.0262.01MR1511422
39. H. Poincaré, Sur une généralisation des fractions continues, C. R. Acad. Sci. Paris 99 (8 déc. 1884) 
40. M.S. Raghunathan, A proof of Oseledets multiplicative ergodic theorem, Israel J. Math. 32 (1979), 356-362 Zbl0415.28013MR571089
41. F. Schweiger, The metrical theory of the Jacobi-Perron algorithm, 334 (1973), Springer Zbl0287.10041MR345925
42. F. Schweiger, A modified Jacobi-Perron algorithm with explicitely given invariant measure, 729 (1979), 199-202, Springer Zbl0411.28022
43. F. Schweiger, The exponent of convergence for the two-dimensional Jacobi-Perron algorithm, Proceedings Conference on Analytic and Elementary Number Theory (1996), 207-213, Vienne Zbl0879.11044
44. A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunoff exponents, Annales de l'Institut Fourier 46 (1996), 325-370 Zbl0853.28007MR1393518
45. H. Kesten, Sums of stationary sequences cannot grow slower than linearly, Proc. Amer. Math. Soc. 49 (1975), 205-211 Zbl0299.28014MR370713
46. T. Fujita, S. Ito, M. Keane, M. Ohtsuki, On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: A corrected proof., Ergodic Theory Dyn. Syst. 16 (1996), 1345-1352 Zbl0868.28008MR1424403

1. Pierre Arnoux, Valérie Berthé, Shunji Ito, Discrete planes, ${\mathbb{Z}}^2$-actions, Jacobi-Perron algorithm and substitutions
2. Nicolas Chevallier, Meilleures approximations diophantiennes simultanées et théorème de Lévy