The Fourier-transform criterion for periodic and divergent trajectories

Let $H:\mathbb{Z}\to\mathbb{Z}$ be a contracting, semi-basic $p$-Hydra map, and let $\widehat{\chi}_H$ be a Fourier transform of its numen. Fourier-transform dynamical criterion. (I) $H$ has finitely many periodic points if and only if

$$\liminf_{|t|_p\to\infty}|\widehat{\chi}_H(t)|_{q_H}=0.$$

(II) If

$$\liminf_{|t|_p\to\infty}|\widehat{\chi}_H(t)|_{q_H}>0,$$

then $H$ has a divergent trajectory. The source presents these as desired conjectural links between Fourier behavior and dynamics; no resolution is supplied.