Under Collatz conjecture the Collatz mapping has no an asymptotic mixing property $\pmod{3}$

Abstract

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.