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UBC Theses and Dissertations

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


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 β‰₯ 1 over 𝔽𝑝. Let X be a variety defined over K, endowed with a dominant rational self-map Ξ¦. We expect that either there exists a variety Y defined over a finite subfield π”½π˜² of 𝔽𝑝 of dimension at least d + 1 and a dominant rational map Ο„: X ‏Y such that Ο„ o 𝝫ᡐ= FΚ³ o Ο„ for some positive integers m and r, where F is the Frobenius endomorphism of Y corresponding to the field π”½π˜², or either there exists Ξ± β‹² X(K) whose orbit under 𝝫 is well-defined and Zariski dense in X, or there exists a non-constant 𝒇 : X ‏ β„™ΒΉ 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.

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