A lower bound on the orbit growth of a regular self-map of affine space

Abstract

We show that if $f:{\mathbb{A}}_{\bar{\mathbb{Q}}}^r\to {\mathbb{A}}_{\bar{\mathbb{Q}}}^r$ is a regular self-map and $P\in {\mathbb{A}}^r(\bar{\mathbb{Q}})$ has ${\mathrm{limsup}}_{n\in \mathbb{N}}\frac{\log h_{\mathrm{aff}}(f^{n}P)}{\log n}<1/r$, where $h_{\text{aff}}$ is the affine Weil height, then $\mathbb{N}$ partitions into a finite set and finitely many full arithmetic progressions, on each of which the coordinates of $f^{n}P$ are polynomials in $n$. In particular, if $(f^{n}P{)}_{n\in \mathbb{N}}$ is a Zariski-dense orbit, then either $n=1$ and $f$ is of the shape $t\mapsto \zeta t+c$, $\zeta \in {\mu }_{\infty }$, or else ${\mathrm{limsup}}_{n\in \mathbb{N}}\frac{\log h_{\mathrm{aff}}(f^{n}P)}{\log n}\ge 1/r$. This inequality is the exponential improvement of the trivial lower bound obtained from counting the points of bounded height in ${\mathbb{A}}^r(K)$.

References in corpus (1)

• A note on a generalization of the Hadamard quotient theorem

Cited by in corpus (2)

• Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian ${\mathbb{Z}}^d$-actions
• Silverman's conjecture for additive polynomial mappings