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Proceedings of the American Mathematical Society

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

 

 

Long periodic orbits of the triangle map


Authors: Manny Scarowsky and Abraham Boyarsky
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
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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}\vert{2^k} \pm 1$. Then there are $ (({\text{p}} - 1) \cdot {p^{N - 1}})/2k$ periodic orbits of $ \tau {\vert _{{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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DOI: https://doi.org/10.1090/S0002-9939-1986-0835874-6
Article copyright: © Copyright 1986 American Mathematical Society