Invariant Subvarieties With Small Dynamical Degree

Abstract

Abstract Let $f:X\to X $ be a dominant self-morphism of an algebraic variety. Consider the set $\Sigma _{f^{\infty }}$ of $f$-periodic subvarieties of small dynamical degree (SDD), 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 SDD and give an optimal upper bound $$\begin{align*} &\sharp P_{f^n}\le d_1(f)^n(1+o(1))\end{align*}$$as $n\to \infty $, where $d_1(f)$ is the 1st dynamic degree. When $X$ is an algebraic group (with $f$ being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound $$\begin{align*} &\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))\end{align*}$$as $n \to \infty $, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.