Dynamics of projective morphisms having identical canonical heights

Abstract

Let ϕ, ψ:ℙN → ℙN be morphisms of degree at least 2 whose canonical heights ĥϕ and ĥψ are identical. We draw various conclusions about the Green functions, Julia sets, and canonical local heights of ϕ and ψ. We use this information to completely characterize ϕ and ψ in the following cases: (i) ϕ and ψ are polynomial maps in one variable; (ii) ϕ is the dth-power map; (iii) ϕ is a Lattès map.