Open Collections

Abstract

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 𝔽𝑝. 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 𝔽𝘲 of 𝔽𝑝 of dimension at least d + 1 and a dominant rational map τ: X ⤏Y such that τ o 𝝫ᵐ= Fʳ o τ for some positive integers m and r, where F is the Frobenius endomorphism of Y corresponding to the field 𝔽𝘲, or either there exists α ⋲ 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 o 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 𝔾ᴺm.