New York Journal of Mathematics
Volume 27 (2021), 1274-1304


Dragos Ghioca and Sina Saleh

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

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Published: August 18, 2021.
Keywords: Zariski dense orbits, Medvedev-Scanlon conjecture, Mordell-Lang theorem in positive characteristic for tori.
Subject: Primary 14K15, Secondary 14G05.

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 bar of dimension at least d + 1 and a dominant rational map τ: X --> Y such that τ ∘ Φm = Fr ∘ 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 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 GmN.


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.

Author information

Dragos Ghioca:
Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2, Canada


Sina Saleh:
Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2, Canada