logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Boris Adamczewski
Symbolic discrepancy and self-similar dynamics
(Discrépance symbolique et dynamiques auto-similaires)
Annales de l'institut Fourier, 54 no. 7 (2004), p. 2201-2234, doi: 10.5802/aif.2079
Article PDF | Reviews MR 2139693 | Zbl 1066.11032
Class. Math.: 11K38, 37A30, 37A45, 37B10, 68R15
Keywords: Discrepancy, substitutions, subshifts, bounded remainder sets, self-similar dynamics

Résumé - Abstract

We consider subshifts arising from primitive substitutions, which are known to be uniquely ergodic dynamical systems. In order to precise this point, we introduce a symbolic notion of discrepancy. We show how the distribution of such a subshift is in part ruled by the spectrum of the incidence matrices associated with the underlying substitution. We also give some applications of these results in connection with the spectral study of substitutive dynamical systems.

Bibliography

[1] B. Adamczewski, “Codages de rotations et phénomènes d'autosimilarité”, J. Théor. Nombres Bordeaux 14 (2002), p. 351-386 Cedram |  MR 2040682 |  Zbl 02184588
[2] B. Adamczewski, “Répartitions des suites $(n\alpha)_{n\in{\bb N}}$ et substitutions”, Acta Arith. 112 (2004), p. 1-22 Article |  MR 2040589 |  Zbl 1060.11043
[3] M.D. Boshernitzan & C.R. Carroll, “An extension of Lagrange's theorem to interval exchange transformations over quadratic fields”, J. Anal. Math. 72 (1997), p. 21-44 Article |  MR 1482988 |  Zbl 0931.28013
[4] J. Brillhart, P. Erdős & P. Morton, “On sums of Rudin-Shapiro coefficients II”, Pacific J. Math. 107 (1983), p. 39-69 Article |  MR 701806 |  Zbl 0469.10034
[5] J. Coquet, “A summation formula related to the binary digits”, Invent. Math. 73 (1983), p. 107-115 Article |  MR 707350 |  Zbl 0528.10006
[6] F.M. Dekking, On the distribution of digits in arithmetic sequences, 1983, p. 1-12 Article |  Zbl 0529.10047
[7] M. Drmota & R.F. Tichy, Sequences, discrepancies and applications, Springer-Verlag, Berlin, 1997  MR 1470456 |  Zbl 0877.11043
[8] J.-M. Dumont & A. Thomas., “Systèmes de numération et fonctions fractales relatifs aux substitutions”, Theoret. Comput. Sci. 65 (1989), p. 153-169 Article |  MR 1020484 |  Zbl 0679.10010
[9] J.-M. Dumont & A. Thomas, “Digital sum problems and substitutions on a finite alphabet”, J. Number Theory 39 (1991), p. 351-366 Article |  MR 1133561 |  Zbl 0736.11007
[10] F. Durand, “A characterization of substitutive sequences using return words”, Discrete Math. 179 (1998), p. 89-101 Article |  MR 1489074 |  Zbl 0895.68087
[11] F. Durand, “Linearly recurrent subshifts have a finite number of non-periodic subshift factors”, Ergodic Theory Dynam. Systems 20 (2000), p. 1061-1078 Article |  MR 1779393 |  Zbl 0965.37013
[12] F. Durand, Combinatorial and dynamical study of substitutions around the theorem of cobham, Kluwer Acad. Publications, 2002, p. 53-94  Zbl 1038.11016
[13] H. Furstenberg, H. Keynes & L. Shapiro, “Prime flows in topological dynamics”, Israel J. Math. 14 (1973), p. 26-38 Article |  MR 321055 |  Zbl 0264.54030
[14] G. Halász, “Remarks on the remainder in Birkhoff's ergodic theorem”, Acta Math. Acad. Sci. Hungar. 28 (1976), p. 389-395 Article |  MR 425076 |  Zbl 0336.28005
[15] C. Holton & L.Q. Zamboni, “Geometric realizations of substitutions”, Bull. Soc. Math. France 126 (1998), p. 149-179 Numdam |  MR 1675970 |  Zbl 0931.11004
[16] H. Kesten, “On a conjecture of Erdős and Szüsz related to uniform distribution $mod 1$”, Acta Arith. 12 (1966/1967), p. 193-212 Article |  MR 209253 |  Zbl 0144.28902
[17] L. Kuipers & H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience, New York, 1974  MR 419394 |  Zbl 0281.10001
[18] D. Lind & B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995  MR 1369092 |  Zbl 00822672
[19] P. Michel, “Stricte ergodicité d'ensembles minimaux de substitution”, C. R. Acad. Sci. Paris Sér. A 278 (1974), p. 811-813  MR 362276 |  Zbl 0274.60028
[20] K. Petersen, “On a series of cosecants related to a problem in ergodic theory”, Compos. Math. 26 (1973), p. 313-317 Numdam |  MR 325927 |  Zbl 0269.10030
[21] M. Queffélec., Substitution dynamical systems - Spectral analysis, Lecture Notes in Mathematics 1294, Springer-Verlag, Berlin, 1987  MR 924156 |  Zbl 0642.28013
[22] G. Rauzy, “Nombres algébriques et substitutions”, Bull. Soc. Math. France 110 (1982), p. 147-178 Numdam |  MR 667748 |  Zbl 0522.10032
[23] G. Rauzy, Sequences defined by iterated morphisms, Springer, 1990, p. 275-286  Zbl 0955.28501
[24] A. Siegel, “Représentation géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot”, Thèse de doctorat de l'Université de la Méditerranée, 2000
[25] N.B. Slater, “Gaps and steps for the sequence $n\theta\mod 1$”, Proc. Cambridge Philos. Soc. 63 (1967), p. 1115-1123 Article |  MR 217019 |  Zbl 0178.04703
[26] B. Solomyak, On the spectral theory of adic transformations, Amer. Math. Soc., 1992, p. 217-230  Zbl 0770.28012
top