TITLE: A question on Collatz's conjecture:proportion of "low flying" orbits
QUESTION [22 upvotes]: 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 ?

REPLY [7 votes]: (Without any intended answer:) Two further pictures.
First, a bit more detailed curve up to $N=100 000$. I supressed the datapoints $T(N)$ at $N$, where the frequencies are below $6$ here to have the curve a bit more sharpened/showing the peaks in the underlying frequencies-list:       
 
And it seems to me, that the actual frequency of occurence of N as maximum of a trajectory gives some more intuition what's going on here. While the above cumulative curve tends to "iron out" any variability when going to the right, the systematic spikes of the frequencies-table here indicate, that such spikes do not wear out in the long run but (likely) even get higher counts (again I left out the datapoints of smaller frequencies in the picture):