Invariant Subvarieties With Small Dynamical Degree

Let f : X → X be a dominant self-morphism of an algebraic variety. Consider the set ∑f8 of f -periodic subvarieties of small dynamical degree (SDD), the subset Sf8 of maximal elements in ∑f8, and the subset Sf of f -invariant elements in Sf8. When X is projective, we prove the finiteness of the set Pf of f -invariant prime divisors with SDD and give an optimal upper bound Pf n = d1(f )n(1 + o(1)) as n→8, where d1(f ) is the 1st dynamic degree. When X is an algebraic group (with f being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound Sf n = d1(f )n dim(X)(1 + o(1)) as n→8, which slightly generalizes a conjecture of S.-W. Zhang for polarized f .