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.References
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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