Very ample polarized self maps extend to projective space

Let $X$ be a projective variety defined over an infinite field, equipped with a line bundle $L$, giving an embedding of $X$ into \mb{P}^m and let $\phi: X \to X$ be a morphism such that $\phi^*L \cong L^{\otimes q}, q\geq 2$. Then there exists an integer $r>0$ extending $\phi^r$ to \mb{P}^m

A dynamical pairing between two rational maps

Given two rational maps $\varphi$ and $\psi$ on \PP^1 of degree at least two, we study a symmetric, nonnegative-real-valued pairing which is closely related to the canonical height functions $h_\varphi$ and $h_\psi$ associated to these maps. Our main results show a strong connection between the value of and the canonical heights of points which are small with respect to at least one of the two maps $\varphi$ and $\psi$. Several necessary and sufficient conditions are given for the vanishing of . We give an explicit upper bound on the difference between the canonical height $h_\psi$ and the standard height $h_\st$ in terms of , where $\sigma(x)=x^2$ denotes the squaring map. The pairing $$ is computed or approximated for several families of rational maps $\psi$