Zariski dense orbits for regular self-maps on split semiabelian varieties

Abstract

Abstract We provide a direct proof of the Medvedev–Scanlon’s conjecture from Medvedev and Scanlon (Ann. Math. Second Series 179(2014), 81–177) regarding Zariski dense orbits under the action of regular self-maps on split semiabelian varieties defined over a field of characteristic $0$ . Besides obtaining significantly easier proofs than the ones previously obtained in Ghioca and Scanlon (Trans. Am. Math. Soc. 369(2017), 447–466; for the case of abelian varieties) and Ghioca and Satriano (Trans. Am. Math. Soc. 371(2019), 6341–6358; for the case of semiabelian varieties), our method allows us to exhibit numerous starting points with Zariski dense orbits, which the methods from Ghioca and Scanlon (Trans. Am. Math. Soc. 369(2017), 447–466) and Ghioca and Satriano (Trans. Am. Math. Soc. 371(2019), 6341–6358) could not provide.