Height Gap Conjectures, $D$-Finiteness, and Weak Dynamical Mordell-Lang

In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form $f({\mathrm{\Phi }}^n(x))$, where $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:X\to X$ and $f\text{\hspace{0.17em}⁣}:X\to {\mathbb{P}}^1$ are rational maps defined over $\bar{\mathbb{Q}}$ and $x\in X(\overline{$\mathbb{Q}$})$ is a point whose forward orbit avoids the indeterminacy loci of $\mathrm{\Phi }$ and $f$. They conjectured that if the sequence is infinite, then $\mathrm{lim\hspace{0.17em}sup}\frac{h(f({\mathrm{\Phi }}^n(x)))}{\log n}>0$. They also made a corresponding conjecture for $\mathrm{lim\hspace{0.17em}inf}$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\mathrm{lim\hspace{0.17em}sup}$ conjecture as well as the $\mathrm{lim\hspace{0.17em}inf}$ conjecture away from a set of density $0$. As applications, we prove results concerning the growth rate of coefficients of $D$-finite power series as well as the Dynamical Mordell-Lang Conjecture up to a set of density $0$.