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 F_p. 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 F_q of F_p bar of dimension at least d + 1 and a dominant rational map τ: X --> Y such that τ ∘ Φ^m = F^r ∘ τ for some positive integers m and r, where F is the Frobenius endomorphism of Y corresponding to the field F_q, or either there exists α in X(K) whose orbit under Φ is well-defined and Zariski dense in X, or there exists a non-constant f:X--> P¹ such that f ∘ Φ = f. 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 G_m^N.

We thank Tom Scanlon who suggested the more precise version of condition (C) from our Conjecture 1.3. We are grateful to the anonymous referee for their useful comments and suggestions, which improved our presentation.