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arXiv · 1905.08589

Arithmetic Convergence of Double-iterated Polynomials

Rin Gotou
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Abstract

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 $b'$.

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Rin Gotou. 2019-05-21. Arithmetic Convergence of Double-iterated Polynomials. https://arxiv.org/abs/1905.08589

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