A. Baragar introduced a canonical vector height on a surface defined over a number field, and showed its existence if has Picard rank two with infinite automorphism group. In another paper, A. Baragar and R. van Lujik performed numerical computation on certain surfaces with Picard rank three, which strongly suggests that, in general, a canonical vector height does not exist. In this note, we prove this last assertion. We compare the set of periodic points of one automorphism with another on certain surfaces.
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Shu Kawaguchi, Canonical vector heights on surfaces – A nonexistence result. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 24 (2013), no. 2, pp. 181–197DOI 10.4171/RLM/651