Finitely balanced sequences and plasticity of 1-dimensional tilings

Sadun, Lorenzo

doi:10.1016/j.topol.2016.01.021

Cited by 6 publications

(4 citation statements)

References 11 publications

(9 reference statements)

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“…However, under the assumptions of Theorem 3.1 (with the balancedness assumption playing a crucial role), our results also apply to the R-action of the associated tiling space (such as investigated e.g. in [CS03]), according to [Sad16].…”

Section: Resultsmentioning

confidence: 88%

Geometry, dynamics, and arithmetic ofS-adic shifts

Berthé¹,

Steiner²,

Thuswaldner³

2019

Annales De l'Institut Fourier

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This paper studies geometric and spectral properties of S-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete spectrum for S-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the S-adic framework. They are applied to families of S-adic shifts generated by Arnoux-Rauzy as well as Brun substitutions. It is shown that almost all of these shifts have pure discrete spectrum. Using S-adic words related to Brun's continued fraction algorithm, we exhibit bounded remainder sets and natural codings for almost all translations on the two-dimensional torus. Due to the lack of self-similarity properties present for substitutive systems we have to develop new proofs to obtain our results in the S-adic setting.

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Section: Resultsmentioning

confidence: 88%

Geometry, dynamics, and arithmetic ofS-adic shifts

Berthé¹,

Steiner²,

Thuswaldner³

2019

Annales De l'Institut Fourier

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“…We convert these questions to questions of cohomology. Since much is already known about the cohomology of tiling spaces ( [4,10,21,30,31,32,35]), this reduces many problems of transport either to a simple look-up or to calculations using well-established techniques [6,5,22,23,24,33,34].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Kelly

Sadun

2020

International Mathematics Research Notices

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Suppose that we have a repetitive and aperiodic tiling T of R n , and two mass distributions f 1 and f 2 on R n , each pattern equivariant with respect to T. Under what circumstances is it possible to do a bounded transport from f 1 to f 2 ? When is it possible to do this transport in a strongly or weakly pattern-equivariant way? We reduce these questions to properties of theČech cohomology of the hull of T, properties that in most common examples are already well-understood.

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“…In the multidimensional framework, balancedness has been considered both for multidimensional words [BT02] and for tilings [Sad16]. This notion has to be compared with homogeneity (related to 0-balance) such as introduced by M. Nivat in [Niv02].…”

Section: Introductionmentioning

confidence: 99%

Balancedness and coboundaries in symbolic systems

Berthé

Cecchi

2019

Theoretical Computer Science

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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.

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Finitely balanced sequences and plasticity of 1-dimensional tilings

Cited by 6 publications

References 11 publications

Geometry, dynamics, and arithmetic ofS-adic shifts

Geometry, dynamics, and arithmetic ofS-adic shifts

Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Balancedness and coboundaries in symbolic systems

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