Dense orbits of rationals

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 $O(x) = \{ {\phi ^i}(x)|i \geqslant 0\} ,\;x \in K$, 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 $k \in K$).