Rational Self-Maps with a Regular Iterate on a Semiabelian Variety

The paper proves Theorem 1.1 via two propositions: (i) if Φ is not regular but Φ^m is regular, then some iterate admits a totally invariant codimension-1 subvariety (Proposition 1.2) , relying on the purity of the indeterminacy locus for maps into group varieties and a Noetherian descent argument ; and (ii) for a dominant regular self-map Ψ admitting such a divisor, there is a nonconstant homomorphism to a semiabelian quotient that is fixed by an iterate (char 0) or Frobenius-semi-conjugate (char p) (Proposition 1.3) , using Pink–Rössler’s structure theory for invariant subvarieties as implemented in Section 5 (e.g., the decomposition (9)) . The candidate solution reproduces exactly this strategy and invokes the same two propositions, then applies the hypotheses to rule out the quotient/Frobenius alternatives, concluding Φ is regular, matching Theorem 1.1(a),(b) . Minor nits: the model states “m≥1” when Proposition 1.2 assumes m>2, but one can replace a given regular iterate by a higher one without changing the argument; similarly the model allows k≥1 where the paper states k>1—both are immaterial. Overall, both are correct and essentially the same proof.