Non-density of points of small arithmetic degrees

Abstract

Given a surjective endomorphism $f:X\to X$ on a projective variety over a number field, one can define the arithmetic degree ${\alpha }_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree ${\alpha }_f(x)$ is strictly smaller than the first dynamical degree ${\delta }_f$ of $f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small arithmetic degree is not Zariski dense, and verify this sAND Conjecture for endomorphisms on projective varieties including surfaces, HyperKähler varieties, abelian varieties, Mori dream spaces, simply connected smooth varieties admitting int-amplified endomorphisms, smooth threefolds admitting int-amplified endomorphisms, and some fibre spaces. We show the equivalence of the sAND Conjecture and another conjecture on the periodic subvarieties of small dynamical degree; we also show the close relations between the sAND Conjecture and the Uniform Boundedness Conjecture of Morton and Silverman on endomorphisms of projective spaces and another long standing conjecture on Uniform Boundedness of torsion points in abelian varieties.

45 pages, minor editing

References in corpus (6)

• Questions on self maps of algebraic varieties
• (Relative) dynamical degrees of rational maps over an algebraic closed field
• On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension
• Invariant subvarieties with small dynamical degree
• The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees
• Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism

Cited by in corpus (4)

• Invariant subvarieties with small dynamical degree
• 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
• Effective Eigendivisors and the Kawaguchi-Silverman Conjecture