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Symbolic discrepancy and self-similar dynamics. (English) Zbl 1066.11032
The author studies symbolic discrepancy for sequences and for subshifts. For infinite sequences (on finite alphabets) this quantity essentially describes how well the sequence sticks to being uniformly distributed with respect to the probability measure naturally associated with this sequence. Without entering into technical details we can comment on one of the main theorems in this paper by saying that the discrepancy of an iterative fixed point of a primitive morphism is governed (for its order of magnitude) by the second eigenvalue of the morphism and its multiplicity, but also – in some precise cases – by an explicit complex number depending on the morphism and on the sequence, that is not invariant by abelianization of the morphism.

##### MSC:
 11K38 Irregularities of distribution, discrepancy 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 37B10 Symbolic dynamics 68R15 Combinatorics on words 11B85 Automata sequences 37A30 Ergodic theorems, spectral theory, Markov operators
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##### References:
 [1] Codages de rotations et phénomènes d’autosimilarité, J. Théor. Nombres Bordeaux, 14, 351-386, (2002) · Zbl 1113.37003 [2] Répartitions des suites $$(nα)_{n∈{\bb N}}$$ et substitutions, Acta Arith., 112, 1-22, (2004) · Zbl 1060.11043 [3] An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields, J. Anal. Math., 72, 21-44, (1997) · Zbl 0931.28013 [4] On sums of rudin-Shapiro coefficients II, Pacific J. Math., 107, 39-69, (1983) · Zbl 0469.10034 [5] A summation formula related to the binary digits, Invent. Math., 73, 107-115, (1983) · Zbl 0528.10006 [6] On the distribution of digits in arithmetic sequences, Seminar on number theory, 1982-1983 (Talence, 1982/1983), exp. no 32, 1-12, (1983), Université Bordeaux I, Talence · Zbl 0529.10047 [7] Sequences, discrepancies and applications, (1997), Springer-Verlag, Berlin · Zbl 0877.11043 [8] Systèmes de numération et fonctions fractales relatifs aux substitutions, Theoret. Comput. Sci., 65, 153-169, (1989) · Zbl 0679.10010 [9] Digital sum problems and substitutions on a finite alphabet, J. Number Theory, 39, 351-366, (1991) · Zbl 0736.11007 [10] A characterization of substitutive sequences using return words, Discrete Math., 179, 89-101, (1998) · Zbl 0895.68087 [11] Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems, 20, 1061-1078, (2000) · Zbl 0965.37013 [12] Combinatorial and dynamical study of substitutions around the theorem of cobham, Dynamics and Randomness, Nonlinear Phenomena and Complex Systems, 53-94, (2002), Kluwer Acad. Publications · Zbl 1038.11016 [13] Prime flows in topological dynamics, Israel J. Math., 14, 26-38, (1973) · Zbl 0264.54030 [14] Remarks on the remainder in Birkhoff’s ergodic theorem, Acta Math. Acad. Sci. Hungar., 28, 389-395, (1976) · Zbl 0336.28005 [15] Geometric realizations of substitutions, Bull. Soc. Math. France, 126, 149-179, (1998) · Zbl 0931.11004 [16] On a conjecture of Erdős and szüsz related to uniform distribution $$mod 1,$$ Acta Arith., 12, 193-212, (19661967) · Zbl 0144.28902 [17] Uniform distribution of sequences, (1974), Wiley-Interscience, New York · Zbl 0281.10001 [18] An introduction to symbolic dynamics and coding, (1995), Cambridge University Press, Cambridge · Zbl 1106.37301 [19] Stricte ergodicité d’ensembles minimaux de substitution, C. R. Acad. Sci. Paris Sér. A, 278, 811-813, (1974) · Zbl 0274.60028 [20] On a series of cosecants related to a problem in ergodic theory, Compos. Math., 26, 313-317, (1973) · Zbl 0269.10030 [21] Substitution dynamical systems - Spectral analysis, 1294, (1987), Springer-Verlag, Berlin · Zbl 0642.28013 [22] Nombres algébriques et substitutions, Bull. Soc. Math. France, 110, 147-178, (1982) · Zbl 0522.10032 [23] Sequences defined by iterated morphisms, Sequences (Naples/Positano, 1988), 275-286, (1990), Springer, New York · Zbl 0955.28501 [24] Représentation géométrique, combinatoire et arithmétique des systèmes substitutifs de type Pisot, (2000) [25] Gaps and steps for the sequence $$nθ\mod 1,$$ Proc. Cambridge Philos. Soc., 63, 1115-1123, (1967) · Zbl 0178.04703 [26] On the spectral theory of adic transformations, Representation theory and dynamical systems, 217-230, (1992), Amer. Math. Soc., Providence, RI · Zbl 0770.28012
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