Tilings of a Riemann surface and cubic Pisot numbers

Enomoto, Fumihiko; Ei, Hiromi; Furukado, Maki; Ito, Shunji

doi:10.32917/hmj/1187916318

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2018

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“…Throughout the paper, we apply our results to a one-parameter family of substitutions defined in Section 3.4, which will constitute our main example. One substitution of this family (see Section 2) was explicitly studied with the same approach in [EEFI07]; a tiling result was obtained and mentioned to hold for the whole family. In Sections 7, we will further relate different definitions for the Rauzy fractals.…”

Section: Introductionmentioning

confidence: 99%

Geometrical Models for a Class of Reducible Pisot Substitutions

Loridant¹,

Minervino²

2018

Discrete Comput Geom

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We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses geometric representations of stepped surfaces and related polygonal tilings, as well as self-replicating and periodic tilings made of Rauzy fractals. We apply our theory to one-parameter family of substitutions. For this family, we analyze and interpret in a new combinatorial way the codings of a domain exchange defined on the associated fractal domains. We deduce that the symbolic dynamical systems associated with this family of substitutions behave dynamically as first returns of toral translations.

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Section: Introductionmentioning

confidence: 99%