"Rocket elements" in bijections $f:\mathbb{N}\to \mathbb{N}$ Let $\mathbb{N}$ denote the set of non-negative integers. If $f:\mathbb{N}\to\mathbb{N}$ is a map, we set for $k\in \mathbb{N}$:


*

*$f^{(0)}(k) = k$, and

*$f^{(n+1)}(k) = f(f^{(n)}(k))$ for all $n\in\mathbb{N}$.


We say $r\in \mathbb{N}$ is a rocket element of $f$ if $f^{(n)}(r) < f^{(n+1)}(r)$ for all $n\in \mathbb{N}$ and denote the set of all rocket elements of $f$ by $\text{Roc}(f)$. 
If $A\subseteq \mathbb{N}$ we define the upper density $\mu^+(A)$ of $A$ by $$\mu^+(A) = \limsup_{n\to\infty}\frac{|A\cap \{0,\ldots n\}|}{n+1}.$$
By $S_\mathbb{N}$ we denote the set of all bijections $f:\mathbb{N}\to \mathbb{N}$.
Question. Is there $f\in S_\mathbb{N}$ such that $\mu^+(\text{Roc}(f)) = 1$?
Bonus question. If not, what is the value of $\sup\{\text{Roc}(f): f\in S_\mathbb{N}\}$?
(Note. Only the question needs to be answered for acceptance, but I am also keen to know about the "bonus question" if the answer cannot be directly inferred from the answer to the question.)
 A: You could also make the condition for $r$ to be a rocket element  be much stricter such as $$f^{(n+1)}(r) \gt f^n(r)^{f^n(r)!}$$ for all $n.$
Pick $A,B$ as before. Define $f$ on $A$ to satisfy this strong rocket condition. You will have a countable number of wildly increasing sequences. Split $B$ into a countable number of countable subsequences in a more or less explicit manner and use them to provide infinite front ends for each increasing sequence. These front ends might be strictly decreasing, but need not be.
A: Choose any infinite set $A\subseteq\mathbb N$ such that $\mu^+(A)=0$. Enumerate both $A$ and $B=\mathbb N\setminus A$ as 
\begin{align*}
A&=\{a_1<a_2< \dots < a_n < \dots\}\\
B&=\{b_1<b_2< \dots < b_n < \dots\}
\end{align*}
Then you can definite the function $f\colon\mathbb N\to\mathbb N$
\begin{align*}
f(b_k)&=b_{k+1}\\
f(a_1)&=b_1\\
f(a_{k+1})&=a_k
\end{align*}
for $k=1,2,\dots$. It is clear that $f$ is a bijection.
For this function $f$, the set of the rocket elements is either $B$ or $B\cup\{a_1\}$ and the asymptotic density of this set is equal to one.
