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Mosc. Math. J., 2006, Volume 6, Number 1, Pages 43–56 (Mi mmj234)  

This article is cited in 2 scientific papers (total in 3 papers)

Statistics of Young diagrams of cycles of dynamical systems for finite tori automorphisms

V. I. Arnol'd

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: A permutation of a set of $N$ elements is decomposing this set into $y$ cycles of lengths $x_s$, defining a partition $N=x_1+…+x_y$. The length $X_1$, the height y and the fullness $\lambda=N/xy$ of the Young diagram $x_1\geq x_2\ge…\ge x_y$ behave for the large random permutation like $x\sim an$, $y\sim b\ln N$, $\lambda\sim c/\ln N$.
The finite 2-torus $M$ is the product $\mathbb Z_m\times\mathbb Z_m$, and its Fibonacci automorphism sends $(u,v)$ to $(2u+v,u+v)$ (mod $m$). This permutation of $N=m^2$ points of the finite torus $M$ defines a peculiar Young diagram, whose behavior (for large $m$) is very different from that of a random permutation of $N$ points.

Key words and phrases: Fibonacci numbers, permutations, symmetric group, projective line, chaos, cat mapping, modular group, randomness generating, Galois field, finite Lobachevsky plane, relativistic de Sitter world.

DOI: https://doi.org/10.17323/1609-4514-2006-6-1-43-56

Full text: http://www.ams.org/.../abst6-1-2006.html
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MSC: 05E10
Received: April 22, 2006
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Citation: V. I. Arnol'd, “Statistics of Young diagrams of cycles of dynamical systems for finite tori automorphisms”, Mosc. Math. J., 6:1 (2006), 43–56

Citation in format AMSBIB
\Bibitem{Arn06}
\by V.~I.~Arnol'd
\paper Statistics of Young diagrams of cycles of dynamical systems for finite tori automorphisms
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 1
\pages 43--56
\mathnet{http://mi.mathnet.ru/mmj234}
\crossref{https://doi.org/10.17323/1609-4514-2006-6-1-43-56}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2265946}
\zmath{https://zbmath.org/?q=an:1124.05096}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208595700003}


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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. “Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. V. I. Arnol'd, “Statistics of the periods of continued fractions for quadratic irrationals”, Izv. Math., 72:1 (2008), 1–34  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. V. I. Arnold, “Permutations”, Russian Math. Surveys, 64:4 (2009), 583–624  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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