A Dynamical Bogomolov Property

A field F is said to have the Bogomolov Property related to a height function h, if h(a) is either zero or bounded from below by a positive constant for all a in F. In this paper we prove that the maximal algebraic extension of a number field K, which is unramified at a place v|p, has the Bogomolov Property related to all canonical heights coming from a Latt\`es map related to a Tate elliptic curve. To prove this algebraical statement we use analytic methods on the related Berkovich spaces.