Let $f$ be a polynomial with integer coefficients such that $f(n)$ positive for any positive integer $n$. We consider diverging sequences $\{ y_n\}$ given by $y_0 = b$ and $y_{ n+1} = f^{ y_n}(a)$ with positive integers $a$ and $b$. We show such a sequence converges in $\widehat{ \mathbb{ Z}}$ and the limit is independent of $b$, if and only if $f$ does not become a permutation of length $p$ on $\mathbb{ Z}/p\mathbb{ Z}$ for any prime number $p$. We also show that $b'$-adic asymptotic approximations of the equation $f^y(a) = y$ holds in $\mathbb{ N}$ for some bases $b'$.
Categories: math.NT math.DS