Abstract
Rauzy introduced a fractal set associated with the two-dimensional toric shift by the vector (𝛽−1, 𝛽−2), where 𝛽 is the real root of the equation 𝛽3 = 𝛽2 + 𝛽 + 1 and showed that this fractal is divided into three fractals that are bounded remainder sets with respect to a given toric shift. The introduced set was named as Rauzy fractal. It obtains many applications in the combinatorics of words, geometry, theory of dynamical systems and number theory. Later, an infinite sequence of tilings of 𝑑 − 1-dimensional Rauzy fractals associated with algebraic Pisot units of the degree 𝑑 into fractal sets of 𝑑 types were introduced. Each subsequent tiling is a subdivision of the previous one. These tilings are closely related to some irrational toric shifts and allowed to obtain new examples of bounded remainder sets for these shifts, and also to get some results on self-similarity of shift orbits. In this paper, we continue the study of generalized Rauzy tilings related to Pisot numbers. A new approach to definition of Rauzy fractals and Rauzy tilings based on expansions of natural numbers on linear recurrence sequences is proposed. This allows to improve the results on the connection of Rauzy tilings and bounded remainder sets and to show that the corresponding estimate of the remainder term is independent on the tiling order. The geometrization theorem for linear recurrence sequences is proved. It states that the natural number has a given endpoint of the greedy expansion on the linear recurrence sequence if and only if the corresponding point of the orbit of toric shift belongs to some set, which is the union of the tiles of the Rauzy tiling. Some number-theoretic applications of this result is obtained. In conclusion, some open problems related to generalized Rauzy tilings are formulated.
Highlights
On the boundary of self-affine tilings generated by Pisot numbers // Journal of Math
Tilings associated with beta-numeration and substitution // Integers: Electronic journal of combinatorial number theory. 2005
A combinatorial approach to products of Pisot substitutions // Ergodic Theory and Dynamical Systems
Summary
В данном разделе мы приводим основные предварительные сведения о фракталах Рози. Подробности могут быть найдены в [7] и [9]–[13]. В [15] доказано, что в этом случае β является единицей Пизо. Из определения числа Пизо вытекает, что |β| > 1, |β(k)| < 1. Здесь и далее черта сверху обозначает замыкание множества. T – линейно связное ограниченное множество и мера границы ∂T равна 0. Далее нам потребуется еще одно определение фрактала Рози T , основанное на комбинаторике слов. В случае рассматривамого нами класса чисел Пизо данный граф содержит d вершин, помеченных числами 0, 1, . Пусть теперь Adm(j) – множество допустимых слов, для которых пути в графе G(β) заканчиваются в вершине j. Каждое множество Rj представляет собой линейно связное ограниченное множество и мера границы ∂Rj равна 0. Различные множества Rj не имеют общих внутренних точек. ⊔ Rd−1, называемое разбиением Рози порядка 0
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