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Yohsuke Matsuzawa, Sheng Meng, Takahiro Shibata, De-Qi Zhang, Guolei Zhong, Invariant Subvarieties With Small Dynamical Degree, International Mathematics Research Notices, Volume 2022, Issue 15, July 2022, Pages 11448–11483, https://doi.org/10.1093/imrn/rnab039
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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$|-invariant prime divisors with SDD and give an optimal upper bound as |$n\to \infty $|, where |$d_1(f)$| is the 1st dynamic degree. When |$X$| is an algebraic group (with |$f$| being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound as |$n \to \infty $|, which slightly generalizes a conjecture of S.-W. Zhang for polarized |$f$|.
$$\begin{align*} &\sharp P_{f^n}\le d_1(f)^n(1+o(1))\end{align*}$$
$$\begin{align*} &\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))\end{align*}$$
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