在Collatz猜想下Collatz映射不具有渐近混合性质$\pmod{3}$
Under Collatz conjecture the Collatz mapping has no an asymptotic mixing property $\pmod{3}$
摘要：
利用$\mathrm{N}$中马氏齐次链和Banach测度的性质，证明了在序列$n$ -证明了所有Collatz序列的第n个坐标等于数$\frac{2}{3}+\frac{(-1)^{n+1}}{3\times 2^{n+1}}.$，并证明了$3m+1$形式的数的一个类似的数字特征等于数$\frac{3}{5}+ \frac{(-1)^{n+1}}{15 \times 2^{2(n-1)}}. $。利用这些公式，证明了在Collatz猜想下，Collatz映射没有渐近混合性质$\pmod{3}$。构造了一个实值函数在所有自然数集合$N$的乘积$N^2$上的例子，使得其重积分（关于$N$中的Banach测度）的等式蕴涵Collatz猜想失败.此外，还证明了Collatz猜想对于超自然数是不成立的.
英文摘要：
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.
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