A Gap Principle for Subvarieties with Finitely Many Periodic Points

TL;DR

This paper establishes a strong gap principle for the occurrence of iterates of a point under a quasi-finite endomorphism intersecting a subvariety with finitely many periodic points, advancing understanding in arithmetic dynamics.

Contribution

It proves a new gap principle for the dynamical Mordell-Lang conjecture in cases where the subvariety has finitely many periodic points and no positive-dimensional periodic subvarieties.

Findings

The set of times when iterates land in the subvariety has large gaps.

The result applies to a special case of the dynamical Mordell-Lang conjecture.

Provides new insights into the distribution of orbits in algebraic dynamics.

Abstract

Let $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the dynamical Mordell-Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{N}~|~f^n(a) \in V \}$ satisfies a strong gap principle.