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

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
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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
Related articles on Google Scholar: Russian articles, English articles

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
2. V. I. Arnol'd, “Statistics of the periods of continued fractions for quadratic irrationals”, Izv. Math., 72:1 (2008), 1–34
3. V. I. Arnold, “Permutations”, Russian Math. Surveys, 64:4 (2009), 583–624