#### Vol. 8, No. 3, 2014

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The algebraic dynamics of generic endomorphisms of $\mathbb{P}^n$

### Najmuddin Fakhruddin

Vol. 8 (2014), No. 3, 587–608
##### Abstract

We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no nontrivial preperiodic subvarieties, any infinite set of preperiodic points is Zariski-dense and any infinite subset of a single orbit is also Zariski-dense, thereby verifying the dynamical “Manin–Mumford” conjecture of Zhang and the dynamical “Mordell–Lang” conjecture of Denis and Ghioca and Tucker in this case.

##### Keywords
generic endomorphisms, projective space
Primary: 37P55
Secondary: 37F10