Symbolic discrepancy and self-similar dynamics

Adamczewski, Boris

doi:10.5802/aif.2079

Cited by 41 publications

(66 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…and this implies the claim, in view of Corollary 3.6. (We should note that (3.1) follows from the work of Adamczewski [1], with an additional argument given in [9]. )…”

Section: Results Of Queffélec and Their Consequencesmentioning

confidence: 98%

Singular substitutions of constant length

Berlinkov¹,

Solomyak²

2018

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We consider primitive aperiodic substitutions of constant length q and prove that, in order to have a Lebesgue component in the spectrum of the associated dynamical system, it is necessary that one of the eigenvalues of the substitution matrix equals √ q in absolute value. The proof is based on results of M. Queffélec, combined with estimates of the local dimension of the spectral measure at zero.

show abstract

“…and this implies the claim, in view of Corollary 3.6. (We should note that (3.1) follows from the work of Adamczewski [1], with an additional argument given in [9]. )…”

Section: Results Of Queffélec and Their Consequencesmentioning

confidence: 98%

Singular substitutions of constant length

Berlinkov¹,

Solomyak²

2018

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…The quantity ∆ v (u) is considered e.g. in [Ada03,Ada04]. One easily checks that u is balanced on the factor v if and only if its discrepancy ∆ v (u) is finite.…”

Section: Balancedness and Discrepancymentioning

confidence: 99%

“…It thus can be determined by the normalized Perron eigenvectors of M σ and M σ 2 . As a consequence, balancedness for factors can be described in terms of the spectrum of the substitution matrices M σ and M σ 2 , which even provide estimates on the symbolic discrepancy and on the balance function B n (u) [Ada03,Ada04].…”

Section: Balancedness and Substitutionsmentioning

confidence: 99%

“…In the case of primitive substitutions, if balancedness is known to be closely related to the eigenvalues of the substitution matrix (see [HZ98,Ada03,Ada04] and Section 2.3), it is not completely characterized in purely linear terms (see Remark 2.6). Note that there is an important literature devoted to the measure of imbalancedness, considered as deviations for Birkhoff sums, for substitutions acting on words or, in the higher-dimensional framework, for tiling substitutions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Balancedness and coboundaries in symbolic systems

Berthé

Cecchi

2019

Theoretical Computer Science

View full text Add to dashboard Cite

This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the numbers of occurrences of a given word in factors of the same length. We focus on two families of words, namely dendric words and words generated by substitutions. The family of dendric words includes Sturmian and Arnoux-Rauzy words, as well as codings of regular interval exchanges. We prove that dendric words are balanced on letters if and only if they are balanced on words. In the substitutive case, we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness.

show abstract

“…Рассмотрим регулярный по Ляпунову сбалансирован-ный марковский компакт X , наибольший показатель Ляпунова которого ра-вен θ 1 …”

Section: сбалансированные регулярные по ляпунову и гиперболические мunclassified