Arithmetic Convergence of Double-iterated Polynomials

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 $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ for any prime number $p$. We also show that $b^{\mathrm{\prime }}$-adic asymptotic approximations of the equation $f^y(a)=y$ holds in $\mathbb{N}$ for some bases $b^{\mathrm{\prime }}$.