Invariant subvarieties with small dynamical degree

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 $Σ_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $Σ_{f^{\infty}}$, 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 $f$.

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