Let $C$ : ${\mathbb N}\longrightarrow {\mathbb N}$ be Collatz's map defined by $C(n) = 3n+1$ if $n$ is odd, and $C(n)=n/2$ if $n$ is even. Then according to Collatz's conjecture, we should have $C^k (n)=1$, for all $n>0$ and $k$ large enough. 
 Assume that the conjecture holds and for $n>0$ define the top of the orbit of $n$ by 
$$
t(n) = \text{max} \{ C^k (n), \ k\geq 0\}
$$


Define 
$$
T(N) = \text{ Card}\{ n\ , \ t(n) \leq N\} /N
$$


Had the assymptotic behaviour of T(N) been studied ?



A question on Collatz's conjecture:proportion of "low flying" orbits
If the sequence $(T(N))\_{N \in \mathbb{N}}$ converges and the limit is
not equal to $0$, this would imply either positive predecessor density for $1$, cf. e.g.


Günther J. Wirsching, [*On the problem of positive predecessor density in $3n+1$ dynamics*](https://www.aimsciences.org/journals/displayArticles.jsp?paperID=235),
Discrete and Continuous Dynamical Systems - Series A (DCDS-A),
Volume 9, Issue 3, 2003, Pages 771 - 787, doi:10.3934/dcds.2003.9.771,


or the existence of nontrivial cycles.


If the sequence converges, it therefore seems hopeless to prove this with
elementary means.


