  C.J. Smyth was among the first to study the spectrum of the Weil height in
the field of all totally real numbers, establishing both lower and upper bounds
for the limit infimum of the height of all totally real integers and
determining isolated values of the height. Later, Bombieri and Zannier
established similar results for totally p-adic numbers and, inspired by work of
Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use
results on equidistribution of points of low height to generalize both
Bogomolov-type results to a wide variety of heights arising in arithmetic
dynamics.
