In 1973 Schinzel proved in  that the standard logarithmic height on the maximal totally real field extension of the rationals is either zero or bounded from below by a positive constant. In this paper we study this property for canonical heights associated to rational functions and the corresponding dynamical system on the affine line.
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Lukas Pottmeyer, Heights and totally real numbers. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 24 (2013), no. 4, pp. 471–483DOI 10.4171/RLM/662