Symbolic discrepancy and self-similar dynamics

[Discrépance symbolique et dynamiques auto-similaires]

Adamczewski, Boris

Annales de l'Institut Fourier, Tome 54 (2004), p. 2201-2234 / Harvested from Numdam

Résumé

Nous considérons des systèmes dynamiques naturellement associés aux substitutions primitives et connus pour être uniquement ergodiques. Afin d'étudier plus précisément cette propriété, nous introduisons différentes notions de discrépance symbolique. Nous montrons comment les propriétés de répartition d'un tel système sont en partie déterminées par les matrices d'incidences associées à la substitution sous-jacente. Nous donnons également certaines applications de ces résultats à l'étude spectrale des systèmes dynamiques substitutifs.

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.

Détail

Publié le : 2004-01-01

MR 2139693 | Zbl 1066.11032

DOI : https://doi.org/10.5802/aif.2079
Classification:  11K38,  37A30,  37A45,  37B10,  68R15
Mots clés: discrépance, substitutions, sous-shifts, ensembles à restes bornés, dynamiques auto-similaires

Bibliographie

[1] B. Adamczewski Codages de rotations et phénomènes d'autosimilarité, J. Théor. Nombres Bordeaux, Tome 14 (2002), pp. 351-386 | Article | Numdam | MR 2040682 | Zbl 02184588

[2] B. Adamczewski Répartitions des suites $(n\alpha)_{n\in{\bb N}}$ et substitutions, Acta Arith., Tome 112 (2004), pp. 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., Tome 72 (1997), pp. 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., Tome 107 (1983), pp. 39-69 | MR 701806 | Zbl 0469.10034

[5] J. Coquet A summation formula related to the binary digits, Invent. Math., Tome 73 (1983), pp. 107-115 | Article | MR 707350 | Zbl 0528.10006

[6] F.M. Dekking On the distribution of digits in arithmetic sequences, Seminar on number theory, 1982-1983 (Talence, 1982/1983), Université Bordeaux I, Talence, Tome exp. no 32 (1983), pp. 1-12 | 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., Tome 65 (1989), pp. 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, Tome 39 (1991), pp. 351-366 | Article | MR 1133561 | Zbl 0736.11007

[10] F. Durand A characterization of substitutive sequences using return words, Discrete Math., Tome 179 (1998), pp. 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, Tome 20 (2000), pp. 1061-1078 | Article | MR 1779393 | Zbl 0965.37013

[12] F. Durand Combinatorial and dynamical study of substitutions around the theorem of cobham, Dynamics and Randomness, Nonlinear Phenomena and Complex Systems, Kluwer Acad. Publications (2002), pp. 53-94 | Zbl 1038.11016

[13] H. Furstenberg; H. Keynes; L. Shapiro Prime flows in topological dynamics, Israel J. Math., Tome 14 (1973), pp. 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., Tome 28 (1976), pp. 389-395 | Article | MR 425076 | Zbl 0336.28005

[15] C. Holton; L.Q. Zamboni Geometric realizations of substitutions, Bull. Soc. Math. France, Tome 126 (1998), pp. 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., Tome 12 (1966/1967), pp. 193-212 | MR 209253 | Zbl 0144.28902

[17] L. Kuipers; H. Niederreiter Uniform distribution of sequences, Wiley-Interscience, New York, Pure and Applied Mathematics (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 1106.37301 | Zbl 00822672

[19] P. Michel Stricte ergodicité d'ensembles minimaux de substitution, C. R. Acad. Sci. Paris Sér. A, Tome 278 (1974), pp. 811-813 | MR 362276 | Zbl 0274.60028

[20] K. Petersen On a series of cosecants related to a problem in ergodic theory, Compos. Math., Tome 26 (1973), pp. 313-317 | Numdam | MR 325927 | Zbl 0269.10030

[21] M. Queffélec. Substitution dynamical systems - Spectral analysis, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1294 (1987) | MR 924156 | Zbl 0642.28013

[22] G. Rauzy Nombres algébriques et substitutions, Bull. Soc. Math. France, Tome 110 (1982), pp. 147-178 | Numdam | MR 667748 | Zbl 0522.10032

[23] G. Rauzy Sequences defined by iterated morphisms, Sequences (Naples/Positano, 1988), Springer, New York (1990), pp. 275-286 | Zbl 0955.28501

[24] A. Siegel Représentation géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot (2000) (Thèse de doctorat de l'Université de la Méditerranée)

[25] N.B. Slater Gaps and steps for the sequence $n\theta\mod 1$, Proc. Cambridge Philos. Soc., Tome 63 (1967), pp. 1115-1123 | Article | MR 217019 | Zbl 0178.04703

[26] B. Solomyak On the spectral theory of adic transformations, Representation theory and dynamical systems, Amer. Math. Soc., Providence, RI (1992), pp. 217-230 | Zbl 0770.28012