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THE AKIYAMA MEAN-MEDIAN MAP HAS UNBOUNDED TRANSIT TIME AND DISCONTINUOUS LIMIT

Part of: Low-dimensional dynamical systems Sequences and sets

Published online by Cambridge University Press:  23 November 2022

JONATHAN HOSEANA Open the ORCID record for JONATHAN HOSEANA [Opens in a new window]
JONATHAN HOSEANA*
Affiliation:
Center for Mathematics and Society, Department of Mathematics, Parahyangan Catholic University, Bandung 40141, Indonesia
*
e-mail: [email protected]
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Abstract

Open conjectures state that, for every $x\in [0,1]$, the orbit $(x_n)_{n=1}^\infty $ of the mean-median recursion

$$ \begin{align*}x_{n+1}=(n+1)\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

with initial data $(x_1,x_2,x_3)=(0,x,1)$, is eventually constant, and that its transit time and limit functions (of x) are unbounded and continuous, respectively. In this paper, we prove that for the slightly modified recursion

$$ \begin{align*}x_{n+1}=n\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.

Keywords

mean-median mapstrong terminating conjecturetransit timelimit functionAkiyama mean-median map

MSC classification

Primary: 37E15: Combinatorial dynamics (types of periodic orbits)
Secondary: 11B75: Other combinatorial number theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 298 - 307
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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