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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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