  We prove that the dynamical system $(\mathbb{N}, 2^{\mathbb{N}}, T, \mu)$,
where $\mu$ is a finite measure equivalent to the counting measure, is
power-bounded in $L^1(\mu)$ if and only if there exists one cycle of the map
$T$ and for any $x \in \mathbb{N}$, there exists $k \in \mathbb{N}$ such that
$T^k(x)$ is in some cycle of the map $T$. This result has immediate
implications for the Collatz Conjecture, and we use it to motivate the study of
number theoretic properties of the inverse image $T^{-1}(x)$ for $x \in
\mathbb{N}$, where $T$ denotes the Collatz map here. We study similar
properties for the related Syracuse maps, comparing them to the Collatz map. We
also analyze some structural properties of the inverse image in relation to
asymptotic density of the set $\{x \in \mathbb{N} \mid \exists k \in
\mathbb{N}: T^k(x) < x\}$.
