К Вопросу О Структуре Самоаффинных Выпуклых Тел

We study self-similar attractors in the space R d , i.e., self-similar compact sets defined by several affine operators with the same linear part. The special case of attractors when the matrix M of the linear part of affine operators and the shifts are integer, is well known in the literature due to many applications in the construction of wavelet and in approximation theory. In this case, if an attractor has measure one, it is called a tile. We classify self-similar attractors and tiles in case when they are either polyhedra or union of finitely many polyhedra. We obtain a complete description of the integer contraction matrices and of the digit sets for tiles-parallelepipeds and for convex tiles in arbitrary dimension. It is proved that on a two-dimensional plane, every polygonal tile (not necessarily convex) must be a parallelogram. Non-trivial examples of multidimensional tiles which are a finite union of polyhedra are given, and in the case d = 1 their complete classification is provided. Applications to orthonormal Haar systems in R d and to integer univariate tiles are considered.

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Compact noncontraction semigroups of affine operators

Войнов¹,

Voynov²,

Протасов³

et al. 2015

Математический сборник

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Компактные несжимающие полугруппы аффинных операторов Исследуются компактные мультипликативные полугруппы аффинных операторов, действующих в конечномерном пространстве. Основной результат утверждает, что либо каждая такая полугруппа является сжимающей, т.е. содержит элементы сколь угодно малой операторной нормы, либо все ее операторы имеют общее инвариантное аффинное подпространство, на котором она является сжимающей. В доказательстве применяются функциональные разностные уравнения со сжатием аргумента. Рассматриваются приложения к задачам о самоаффинных разбиениях выпуклых множеств, к описанию конечных аффинных полугрупп, а также к доказательству критерия примитивности семейства неотрицательных матриц. Библиография: 32 названия.

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К Вопросу О Структуре Самоаффинных Выпуклых Тел

Cited by 4 publications

References 10 publications

Compact noncontraction semigroups of affine operators

Compact noncontraction semigroups of affine operators

Simple tiles and attractors

Compact noncontraction semigroups of affine operators

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