A rational orbit that's provably dense in the reals? Iterating the map $\ \ x\ \mapsto\  x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational whose orbit is provably dense in $\mathbb{R}$?
NB: "Rational mapping" here means simply a function from rationals to rationals, not the definition in algebraic geometry.

EDIT: Does the following approach work? ...
The answer to another posted question proves that with 
$$f(x)=\dfrac1{2 \lfloor x \rfloor -x+1}$$the rational orbit $$1,\ f(1),\ f(f(1)),\ ...,\ f^n(1),\ ...$$
is the Calkin-Wilf sequence containing every positive rational exactly once, and is therefore dense in $\mathbb{R^+}$. 
Question: Can it be shown that in this Calkin-Wilf sequence the even-index rationals alone are dense in $\mathbb{R^+}$, and likewise for the odd-index rationals? 
If so, then, noting that $f(0)=1$, we can obtain a rational orbit that's provably dense in the whole of $\mathbb{R}$ by simply taking 
$$0,\ g(0),\ g(g(0)),\ ...,\ g^n(0),\ ...$$
with 
$$g(x) =  
\begin{cases} 
-f(x)  & \text{if }x\ge 0 \\
f(-x) & \text{if }x<0 
\end{cases}$$
so
$$g^n(0) =  
\begin{cases} 
f^n(0)  & \text{if }n\text{ is even} \\
-f^n(0) & \text{if }n\text{ is odd}. 
\end{cases}$$
 A: Start from the theorem that for any irrational $\mu \in (0,1)$ the set 
$$
\{2^k \mu\} : k \in \Bbb{N}
$$
where (following Knuth's Concrete Mathematics) the notation $\{x\}$ means the fractional part of $x$ ($ \equiv x-\lfloor x \rfloor$) is dense in $(0,1)$.
Now look at the map 
$$
x \mapsto \frac{x - \frac{1}{x}}{2} $$
with some initial point $x_0$.
The equation for this map superficially looks a lot like the example you give.  But that $2$ in the denominator does a couple of things:


*

*$x_n$ is the $n$-th guess you would obtain if you were naively to try to find $\sqrt{-1}$ using Newton's algorithm with a starting guess of $x_0$.

*We can express the value of $x_n$ in closed form (I attribute this to Carl Bender in a course at MIT in 1973 or so, but of course it may have been known much earlier):
$$ x_n = \cot \left( 2^n \cot^{-1}x_0 \right)$$
But as long as $\cot{-1}x_0$ is not a rational multiple of $2\pi$, by the theorem stated above the set 
$$
\{2^k \cot^{-1}x_0\} : k \in \Bbb{N}
$$
is dense in $(0,2\pi)$.
So the set of values of $x_k$ is the set of cotangents of a dense set on the circle; in turn, the set of values of $x_k$ is dense on the real line.  An example is an initial value of $x_0 = 2$.
