Increasing-decreasing patterns in the iteration of an arithmetic function

Abstract

Let $Ω$ be a set of positive integers and let $f:Ω\rightarrow Ω$ be an arithmetic function. Let $V = (v_i)_{i=1}^n$ be a finite sequence of positive integers. An integer $m \in Ω$ has \textit{increasing-decreasing pattern} $V$ with respect to $f$ if, for all odd integers $i \in \{1,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) < f^{v_1+ \cdots + v_{i-1}+1}(m) < \cdots < f^{v_1+ \cdots + v_{i-1}+v_{i}}(m) \] and, for all even integers $i \in \{2,\ldots, n\}$, \[ f^{v_1+ \cdots + v_{i-1}}(m) > f^{v_1+ \cdots +v_{i-1}+1}(m) > \cdots > f^{v_1+ \cdots +v_{i-1}+v_i}(m). \] The arithmetic function $f$ is \textit{wildly increasing-decreasing} if, for every finite sequence $V$ of positive integers, there exists an integer $m \in Ω$ such that $m$ has increasing-decreasing pattern $V$ with respect to $f$. This paper gives a proof that the Syracuse function is wildly increasing-decreasing.