Decidability Problems for Self-induced Systems Generated by a Substitution

International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems

The structure of invertible substitutions on a three-letter alphabet

AbstractWe study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as Iw∘σ1∘σ2∘⋯∘σk, where Iw is the inner automorphism associated with w, and σj∈S for 1⩽j⩽k. As a consequence, M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices

Обобщённые разбиения Рози и множества ограниченного остатка

Rauzy introduced a fractal set associted with the toric shift by the vector (β−1, β−2), where β is the real root of the equation β3 = β2 + β + 1. He show that this fractal can be partitioned into three fractal sets that are bounded remaider sets with respect to the considered toric shift. Later, the introduced set was named as the Rauzy fractal. Further, many generalizations of Rauzy fractal are discovered. There are many applications of the generalized Rauzy fractals to problems in number theory, dynamical systems and combinatorics.Zhuravlev propose an infinite sequence of tilings of the original Rauzy fractal and show that these tilings also consist of bounded remainder sets. In this paper we consider the problem of constructing similar tilings for the generalized Rauzy fractals associated with algebraic Pisot units.We introduce an infinite sequence of tilings of the d−1-dimensional Rauzy fractals associated with the algebraic Pisot units of the degree d into fractal sets of d types. Each subsequent tiling is a subdivision of the previous one. Some results describing the self-similarity properties of the introduced tilings are proved.Also, it is proved that the introduced tilings are so called generalized exchanding tilings with respect to some toric shift. In particular, the action of this shift on the tiling is reduced to exchanging of d central tiles. As a corollary, we obtain that the Rauzy tiling of an arbitrary order consist of bounded remainder sets with respect to the considered toric shift.In addition, some self-similarity property of the orbit of considered toric shift is established.Рози ввел фрактальное множество, связанное со сдвигом двумерного тора на вектор (β−1, β−2), где β – действительный корень уравнения β3 = β2+ β +1 и показал, что данный фрактал разбивается на три фрактала, являющихся множествами ограниченного остатка относительно данного сдвига тора. Введенное множество получило название фрактала Рози. В дальнейшем были введены многочисленные обобщения фракталов Рози, нашедшие применения в целом ряде задач теории чисел, теории динамических систем и комбинаторики.Журавлев ввел бесконечную последовательность разбиений исходного фрактала Рози на фрактальные множества и показал, что они также состоят из множеств ограниченного остатка. В настоящей работе рассматривается задача о построении обобщения таких разбиений для фракталов Рози, связанных с алгебраическими единицами Пизо.В работе введена бесконечная последовательность разбиений d − 1-мерных фракталов Рози, связанных с алгебраическими единицами Пизо степени d, на фрактальные множества d типов. Каждое следующее разбиение последовательности является подразбиением предыдущего. Доказан ряд свойств, описывающих самоподобие введенных разбиений.Показано, что введенные разбиения являются так называемыми обобщенными перекладывающимися разбиениями относительно некоторого сдвига тора. В частности, действие данного сдвига на разбиении сводится к перекладыванию d центральных фигур разбиения. В качестве следствия получено, что разбиение Рози произвольного порядка состоит из множеств ограниченного остатка относительно рассматриваемого сдвига тора.Также доказано, что орбита рассматриваемого сдвига тора обладает свойством самоподобия

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits

Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions

Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions

The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms. The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrational examples by using multidimensional continued fractions. We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions

Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions