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.)