Please use this identifier to cite or link to this item:
Title: Non-density of points of small arithmetic degrees
Authors: Matsuzawa, Yohsuke
Meng, Sheng 
Shibata, Takahiro 
Zhang, De-Qi 
Keywords: math.AG
Primary: 37P55, Secondary: 14G05, 37B40
Issue Date: 25-Feb-2020
Citation: Matsuzawa, Yohsuke, Meng, Sheng, Shibata, Takahiro, Zhang, De-Qi (2020-02-25). Non-density of points of small arithmetic degrees. ScholarBank@NUS Repository.
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, HyperK\"ahler varieties, abelian varieties, Mori dream spaces, simply connected smooth varieties admitting int-amplified endomorphisms, smooth threefolds admitting int-amplified endomorphisms, and some fiber spaces. We also show close relations between our 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.
Appears in Collections:Staff Publications

Show full item record
Files in This Item:
File Description SizeFormatAccess SettingsVersion 
2002.10976v1.pdf509.07 kBAdobe PDF



Google ScholarTM


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.