chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2021-22-2-202-235cheb-991Research ArticleСтатьиArticleКомбинаторика слов, фактординамика и нормальные формыCombinatorics on words, facrordynamics and normal formsРешетниковАндреевич ИванReshetnikovIvan Andreevich

аспирант

graduate student

reshetnikov.ivan@phystech.eduМосковский физико-технический институтРоссияMoscow Institute of Physics and TechnologyRussian Federation202101062021222202235Copyright © Решетников А.И., 20212021Решетников А.И.Reshetnikov I.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/991

Методы символической динамики играют существенную роль в изучении комбинаторных свойств слов, задачах теории чисел и теории динамических систем. Работа посвящена задачам комбинаторике слов, её приложениям в алебре и динамических системах. В разделе 2.1 рассматривается одномерный случай на ключевом примере слов Штурма. Даётся доказательство критерия подстановочности палиндромов Штурма с помощью индукции Рози, рассматривается случай одномерной фактординамики. В разделе 2.2 рассматривается сдвиг тора и фрактал Рози, порождающий слово Трибоначчи. Рассказывается связь периодичности схем Рози и подстановочности слова, порождённого этой системой. Приводится реализация слова Трибоначчи через перекладывание отрезков. Намечается подход к гипотезе Пизо. В разделе 2.3 говорится об унипотентных преобразованиях тора и бильярдах в многоугольниках.В главе 3 рассказывается о нормальных формах и росте групп и алгебр. Глава 4 посвящена графам Рози, базисам Гребнера и ко-росту, а также алгебраическим применениям. В разделе 4.1 говорится о результатах в комбинаторике полилинейных слов, разбитой В. Н. Латышевым и поставленных им проблемах. В параграфе 4.2 говорится о конечноопределённых объектах и проблемах контроля над определяющими их соотношениями. В разделе 4.3 описываются некоторые мономиальные алгебры в терминах равномерно рекуррентных слов.Глава 5 посвящена проблеме о высоте и нормальным формам.

Methods of symbolic dynamics play an essential role in the study of combinatorial properties of words, problems in number theory and the theory of dynamical systems. The paper is devotedto the problems of combinatorics on words, its applications in algebra and dynamical systems.Section 2.1 considers the one-dimensional case using the key example of Sturm’s words. The proof of the criterion for substitutionality of Sturm palindromes using the Rauzy induction is given, the case of one-dimensional facordynamics is considered. Section 2.2 discusses the shift of the torus and the Rauzy fractal that generates the word Tribonacci. The relationship between the periodicity of Rauzy’s schemes and the substitutionality of the word generated by this system is discussed. The implementation of the word Tribonacci through the rearrangement of line segments is given. An approach to the Pisot hypothesis is outlined. Section 2.3 talks aboutunipotent torus transformations and billiards in polygons.Chapter 3 talks about normal forms and the growth of groups and algebras. Chapter 4 is devoted to Rosie graphs, Gr¨obner bases and co-growth, and algebraic applications. Section 4.1discusses the results in the combinatorics of multilinear words developed by V. N. Latyshev and the problems he posed. Section 4.2 talks about finitely defined objects and the problems ofcontrolling the relationships that define them. Section 4.3 describes some monomial algebras in terms of uniformly recurrent words.Chapter 5 deals with the problem of height and normal forms.

комбинаторика словслова Штурмафактординамикафракталы Ро- зиперекладывания отрезковсимволическая динамикатеорема Ширшова о высотепочти периодическая последовательностьтеорема о высотенепериодические мозаикиDOLL- системытеоремы типа теоремы Вершика-Лившицасхемы Розиопределимость в струк- турахбазис Гребнера-Ширшова𝑛-разбиваемостьтеоремы Дилуорсапроблемы бернсай- довского типаcombinatorics of wordsSturm wordsfactordynamicsRauzy fractalsinterval exchangingssymbolic dynamicsShirshov’s height theoremalmost periodic sequenceheight theoremnon-periodic mosaicsDOLL-systemsVershik-Livshitz type theoremsRauzy schemesdefinability in structuresthe Gr¨obner-Shirshov basi𝑛-splitabilityDilworth’s theoremsBurnside type problemsReferencesС.И. Адян. Проблема Бернсайда и тождества в группах // М., Наука, 1975.Adian S.I. The Burnside problem and identities in groups, (in Russian) Izdat. Nauka’, Moscow, 1975. 335 pp.Адян С. И., Разборов А. А. 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The authors declare that there are no conflicts of interest present.