The correspondence principle for divergent points

Let $p$ be a prime, let $H:\mathbb{Z}\to\mathbb{Z}$ be a suitable $p$-adic Hydra map satisfying prerequisites such as those in the source's theorem, and let $\chi_H:\mathbb{Z}_p\to\mathbb{Z}$ be its numen. Let $\mathbb{Z}_p\setminus\mathbb{Q}$ denote the irrational $p$-adic integers. Correspondence principle for divergent points. An integer $x\in\mathbb{Z}$ belongs to a divergent trajectory under $H$ if and only if there exists $\mathfrak z\in\mathbb{Z}_p\setminus\mathbb{Q}$ such that $\chi_H(\mathfrak z)\in\mathbb{Z}$. This is proposed as the divergent-point analogue of the periodic-point correspondence principle; the source says that a constructive method for producing such $\mathfrak z$ is still lacking.