 

Dragos Ghioca and
Sina Saleh
Zariski dense orbits for regular selfmaps of tori in positive characteristic
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Published: 
August 18, 2021. 
Keywords: 
Zariski dense orbits, MedvedevScanlon conjecture,
MordellLang theorem in positive characteristic for tori. 
Subject: 
Primary 14K15, Secondary 14G05. 


Abstract
We formulate a variant in characteristic p of the Zariski dense orbit conjecture previously posed by Zhang, MedvedevScanlon and AmerikCampana for rational selfmaps 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 selfmap 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 welldefined and Zariski dense in X, or there exists a nonconstant f:X> P^{1} 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 selfmaps on
G_{m}^{N}.


Acknowledgements
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
dghioca@math.ubc.ca
Sina Saleh:
Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2, Canada
sinas@math.ubc.ca

