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Funktsional. Anal. i Prilozhen., 2003, Volume 37, Issue 3, Pages 20–35 (Mi faa155)  

This article is cited in 11 scientific papers (total in 12 papers)

The Topology of Algebra: Combinatorics of Squaring

V. I. Arnol'dab

a Université Paris-Dauphine
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We study the graph each of whose edges connects an element of a given ring with the square of itself. For a finite commutative group (e.g., for the multiplicative group of coprime residue classes modulo a positive integer), we describe this graph explicitly: each of its connected components is an oriented attracting cycle equipped with identical $2^k$-vertex rooted trees of special form whose roots reside on the cycle. We also compute the graphs of permutation groups on not too many elements and of the subgroups of even permutations; the connected components of these graphs are also uniformly equipped cycles.

Keywords: Euler function, Fermat's little theorem, quadratic residues, geometric series, attractor, tree, permutation, Young diagram

DOI: https://doi.org/10.4213/faa155

Full text: PDF file (291 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2003, 37:3, 177–190

Bibliographic databases:

Document Type: Article
UDC: 51+515+512+519.1+511+517.938
Received: 08.01.2003

Citation: V. I. Arnol'd, “The Topology of Algebra: Combinatorics of Squaring”, Funktsional. Anal. i Prilozhen., 37:3 (2003), 20–35; Funct. Anal. Appl., 37:3 (2003), 177–190

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Arnol'd, “Topology and statistics of formulae of arithmetics”, Russian Math. Surveys, 58:4 (2003), 637–664  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. A. V. Zarelua, “On matrix analogs of Fermat's little theorem”, Math. Notes, 79:5 (2006), 783–796  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. “Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. Shparlinski, IE, “On some dynamical systems in finite fields and residue rings”, Discrete and Continuous Dynamical Systems, 17:4 (2007), 901  crossref  mathscinet  zmath  isi  scopus
    5. A. V. Zarelua, “On Congruences for the Traces of Powers of Some Matrices”, Proc. Steklov Inst. Math., 263 (2008), 78–98  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. Mazur M., Petrenko B.V., “Generalizations of Arnold's version of Euler's theorem for matrices”, Jpn J Math, 5:2 (2010), 183–189  crossref  mathscinet  zmath  isi  scopus
    7. Ramos A.D., Toom A., “Trajectories in Random Monads”, J Stat Phys, 142:1 (2011), 201–219  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Ramos A.D., Toom A., “Phase Transitions in the Dynamics of Slow Random Monads”, J Stat Phys, 145:5 (2011), 1324–1342  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. R. S. Ismagilov, “A Formula for the Spectra of Differential Operators on Graphs”, Funct. Anal. Appl., 46:2 (2012), 94–99  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. Ramos A.D., Toom A., “Moments and Distributions of Trajectories in Slow Random Monads”, J. Stat. Phys., 147:3 (2012), 623–633  crossref  mathscinet  zmath  adsnasa  isi  scopus
    11. Canovas Pena J.S., Linero Bas A., Soler Lopez G., “a Converse Result Concerning the Periodic Structure of Commuting Affine Circle Maps”, J. Nonlinear Sci. Appl., 9:7 (2016), 5041–5060  mathscinet  zmath  isi
    12. V. S. Kalnitskii, A. N. Petrov, “Lokalnye gladkie sopryazheniya endomorfizmov Frobeniusa”, Geometriya i topologiya. 13, Zap. nauchn. sem. POMI, 476, POMI, SPb., 2018, 111–124  mathnet
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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