论文标题
具有小动力学程度的不变亚体型
Invariant subvarieties with small dynamical degree
论文作者
论文摘要
令$ f：x \ x $是代数封闭的特征零字段，是代数变体的主要自变态。 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在$ s_ {f^{\ infty}} $中。当$ x $投影时，我们证明了$ f $ f $ f $ invariant的$ p_f $的有限性，具有较小的动力学程度，并给出了最佳的（基数）$$ \ sharp p_ {f^n} \ le d_1（f）^n（1+o（1+o（1+o（1+o）， $ f $。当$ x $是一个代数群（$ f $是一个差异的翻译），或（不一定是完整的）复（$ f $稳定大圆环）时，我们给出了最佳的上限$ \ \ \ sharp s_ {f^n} \ le d_1（f） \ infty $，略微概括了S.-W的猜想。 Zhang for两极分化$ f $。
Let $f:X\to X $ be a dominant self-morphism of an algebraic variety over an algebraically closed field of characteristic zero. We consider the set $Σ_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $Σ_{f^{\infty}}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty}}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with small dynamical degree, and give an optimal upper bound (of cardinality) $$\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 $f$.
