# Height Bounds and Preperiodic Points for Families of Jointly Regular Affine Maps

@article{Silverman2006HeightBA, title={Height Bounds and Preperiodic Points for Families of Jointly Regular Affine Maps}, author={Joseph H. Silverman}, journal={Pure and Applied Mathematics Quarterly}, year={2006}, volume={2}, pages={135-145} }

h ( φ(P ) ) = d · h(P ) + O(1) for all P ∈ P (K̄) combined with the fact that there are only finitely many K-rational points of bounded height leads immediately to a proof of Northcott’s Theorem [18] stating that φ has only finitely many K-rational preperiodic points. The situation is more complicated if φ : PN → PN is only required to be a rational map. An initial difficulty arises because there may be orbits Oφ(P ) that “terminate” because some iterate φn(P ) arrives at a point where φ is not…

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