Non-density of Points of Small Arithmetic Degrees

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