Invariant subvarieties with small dynamical degree

TL;DR

This paper studies the structure and bounds of invariant subvarieties with small dynamical degree under dominant self-morphisms, providing finiteness results and optimal bounds in various algebraic settings.

Contribution

It establishes finiteness of invariant prime divisors with small dynamical degree and derives optimal bounds for their number under iteration, generalizing previous conjectures.

Findings

Finiteness of the set of invariant prime divisors with small dynamical degree.

Optimal upper bounds for the number of invariant subvarieties under iteration.

Generalization of Zhang's conjecture for polarized maps on algebraic groups and toric varieties.

Abstract

Let $f:X\to X $ be a dominant self-morphism of an algebraic variety over an algebraically closed field of characteristic zero. We consider the set $\Sigma_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $\Sigma_{f^{\infty}}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty}}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with small dynamical degree, and give an optimal upper bound (of cardinality) $$\sharp P_{f^n}\le d_1(f)^n(1+o(1))$$ as $n\to \infty$, where $d_1(f)$ is the first dynamic degree of $f$. When $X$ is an algebraic group (with $f$ being a translation of an isogeny), or a (not necessarily complete) toric variety (with $f$ stabilizing the big torus), we give an optimal upper bound $$\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))$$ as $n \to \infty$, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.