Zariski dense orbits for regular self-maps of tori in positive characteristic

We formulate a variant in characteristic p of the Zariski dense orbit conjecture previously posed by Zhang, Medvedev-Scanlon and Amerik-Campana for rational self-maps of varieties defined over fields of characteristic 0. So, in our setting, let K be an algebraically closed field, which has transcendence degree d ≥ 1 over . Let X be a variety defined over K, endowed with a dominant rational self-map Φ. We expect that either there exists a variety Y defined over a finite subfield of of dimension at least d + 1 and a dominant rational map τ: X ⤏Y such that τ o ᵐ= Fʳ o τ for some positive integers m and r, where F is the Frobenius endomorphism of Y corresponding to the field , or either there exists α ⋲ X(K) whose orbit under is well-defined and Zariski dense in X, or there exists a non-constant : X ⤏ ℙ¹ such that o = . We explain why the new condition in our conjecture is necessary due to the presence of the Frobenius endomorphism in case X is isotrivial. Then we prove our conjecture for all regular self-maps on ᴺm.Science, Faculty ofMathematics, Department ofGraduat