  By using properties of Markov homogeneous chains and Banach measure in
$\mathrm{N}$, it is proved that a relative frequency of even numbers in the
sequence of $n$-th coordinates of all Collatz sequences is equal to the number
$\frac{2}{3}+\frac{(-1)^{n+1}}{3\times 2^{n+1}}.$ It is shown also that an
analogous numerical characteristic for numbers of the form $3m+1$ is equal to
the number $\frac{3}{5}+ \frac{(-1)^{n+1}}{15 \times 2^{2(n-1)}}. $ By using
these formulas it is proved that under Collatz conjecture the Collatz mapping
has no an asymptotic mixing property $\pmod{3}$. It is constructed also an
example of a real-valued function on the cartesian product $N^2$ of the set of
all natural numbers $N$ such that an equality its repeated integrals (with
respect to Banach measure in $N$) implies that Collatz conjecture fails. In
addition, it is demonstrated that Collatz conjecture fails for supernatural
numbers.
