Invariant subvarieties with small dynamical degree

Abstract

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 ${\Sigma }_{f^{\infty }}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, 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 small dynamical degree, and give an optimal upper bound (of cardinality) $$♯P_{f^n}\le d_1(f{)}^n(1+o(1))$$ 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 $$♯S_{f^n}\le d_1(f{)}^{n\cdot \dim (X)}(1+o(1))$$ as $n\to \infty$, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.

Minor revision, 31 pages, International Mathematics Research Notices (to appear)

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• Pulling Back Cohomology Classes and Dynamical Degrees Of Monomial Maps
• Non-density of points of small arithmetic degrees
• The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees
• Rigidity of rationally connected smooth projective varieties from dynamical viewpoints
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Cited by in corpus (5)

• Non-density of points of small arithmetic degrees
• Canonical heights for abelian group actions of maximal dynamical rank
• Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
• Zariski density of points with maximal arithmetic degree for surfaces
• Potential density of projective varieties having an int-amplified endomorphism