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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Dense orbits of rationals


Author: Michael D. Boshernitzan
Journal: Proc. Amer. Math. Soc. 117 (1993), 1201-1203
MSC: Primary 58F08
MathSciNet review: 1134622
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Abstract: Let $ \mathbb{Q}$ denote the rational points of the interval $ K = [0,1)$. We construct a one-to-one piecewise linear map $ \phi :K \to K$ which has the following properties:

(1) for any $ x \in K,\phi (x) \in \mathbb{Q}$ if and only if $ x \in \mathbb{Q}$;

(2) all the orbits $ O(x) = \{ {\phi ^i}(x)\vert i \geqslant 0\} ,\;x \in K$, are dense in $ K$;

(3) $ \phi $ is an automorphism of the unit circle $ K = [0,1) = \mathbb{R}/\mathbb{Z}$.

This example is motivated by a question of Friedman who was interested, because of an application to logic (Dynamic Recursion Theory), in an example of a piecewise polynomial map $ \phi :K \to K$ having an orbit $ O(k)$ that is dense in $ K$ and lies in $ \mathbb{Q}$ (for some $ k \in K$).


References [Enhancements On Off] (What's this?)

  • [H] Michael Herman, Sur la conjugason différentiable des diffeomorphisms du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5-253.
  • [R] Liming Ren, personal communication, May 1991.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1134622-6
PII: S 0002-9939(1993)1134622-6
Article copyright: © Copyright 1993 American Mathematical Society