  The $3x+1$ map $T$ is defined on the $2$-adic integers $\mathbb{Z}_2$ by
$T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. It is still unproved
that under iteration of $T$ the trajectory of any rational $2$-adic integer is
eventually cyclic. A $2$-adic integer is rational if and only if its
representation with $1$'s and $0$'s is eventually periodic. We prove that the
$3x+1$ conjugacy $\Phi$ maps aperiodic $v\in\mathbb{Z}_2$ onto aperiodic
$2$-adic integers provided that
$\underline{\lim}\;(\frac{h}{\ell})_{\ell=1}^{\infty} > \frac{\ln(2)}{\ln(3)}$
where $h$ is the number of $1$'s in the first $\ell$ digits of $v$ with the
following constraint: if there is a rational $2$-adic integer with a non-cyclic
trajectory, then necessarily
$\underline{\lim}\;(\frac{h}{\ell})_{\ell=1}^{\infty}=\frac{\ln(2)}{\ln(3)}$.
We study $\Phi$ as an infinite series in $\mathbb{R}$ and obtain negative
irrational numbers for which we compute their aperiodic $2$-adic expansion. We
find prominent behaviors of the orbit of $x$ taking Sturmian words as parity
vector. We also found amazing results of the terms of $\Phi$ in $\mathbb{R}$.
We define the $\ell$'th iterate of $T$ for $\ell\rightarrow \infty$ in the ring
of $3$-adic integers and obtain positive irrational numbers for which we
compute their aperiodic $3$-adic expansion.
