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

Abstract

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 $f(\hat{I}{¦}^n(x))$, where $\hat{I}¦:X\to X$ and $f:X\to {\mathbb{P}}^1$ are rational maps defined over $\overline{\mathbb{Q}}$ and $x\in X(\overline{\mathbb{Q}})$ is a point whose forward orbit avoids the indeterminacy loci of $\hat{I}¦$ and $f$. They conjectured that if the sequence is infinite, then $\mathrm{limsup}\frac{h(f(\hat{I}{¦}^n(x)))}{\log n}>0$. They also made a corresponding conjecture for $\mathrm{liminf}$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\mathrm{limsup}$ conjecture as well as the $\mathrm{liminf}$ 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 $0$.