Invariant Subvarieties With Small Dynamical Degree

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