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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Long periodic orbits of the triangle map
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by Manny Scarowsky and Abraham Boyarsky PDF
Proc. Amer. Math. Soc. 97 (1986), 247-254 Request permission

Abstract:

Let $\tau :[0,1] \to [0,1]$ be defined by $\tau (x) = 2x$ on $[0,1/2]$ and $\tau (x) = 2(1 - x)$ on $[1/2,1]$. We consider $\tau$ restricted to the domain ${D_N} = \{ 2a/{p^N},N \geqslant 1,0 \leqslant 2a \leqslant {p^N},(a,p) = 1\}$ where $p$ is any odd prime. Let $k \geqslant 1$ be the minimum integer such that ${p^N}|{2^k} \pm 1$. Then there are $(({\text {p}} - 1) \cdot {p^{N - 1}})/2k$ periodic orbits of $\tau {|_{{D_N}}}$, having equal length, and there are $k$ points in each orbit. Furthermore, the proportion of points in any of these periodic orbits which lie in an interval $(c,d)$ approaches $d - c$ as ${p^{N - 1}} \to \infty$. An application to the irreducibility of certain nonnegative matrices is given.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 97 (1986), 247-254
  • MSC: Primary 28D05; Secondary 15A36, 58F08, 58F22
  • DOI: https://doi.org/10.1090/S0002-9939-1986-0835874-6
  • MathSciNet review: 835874