Increasing-decreasing patterns in the iteration of an arithmetic function

Let $\mathrm{\Omega }$ be a set of positive integers and let $f:\mathrm{\Omega }\to \mathrm{\Omega }$ be an arithmetic function. Let $V=(v_i{)}_{i=1}^n$ be a finite sequence of positive integers. An integer $m\in \mathrm{\Omega }$ has $\mathit{\text{increasing-decreasing}}\mathit{\text{pattern}}$ $V$ with respect to $f$ if, for all odd integers $i\in \{1,\dots ,n\}$, $$f^{v_1+\cdots +v_{i-1}}(m)<f^{v_1+\cdots +v_{i-1}+1}(m)<\cdots <f^{v_1+\cdots +v_{i-1}+v_i}(m)$$ and, for all even integers $i\in \{2,\dots ,n\}$, $$f^{v_1+\cdots +v_{i-1}}(m)>f^{v_1+\cdots +v_{i-1}+1}(m)>\cdots >f^{v_1+\cdots +v_{i-1}+v_i}(m)\mathrm{.}$$ The arithmetic function $f$ is $\mathit{\text{wildly}}\mathit{\text{increasing-decreasing}}$ if, for every finite sequence $V$ of positive integers, there exists an integer $m\in \mathrm{\Omega }$ 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.