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

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.

generic endomorphisms, projective space
Mathematical Subject Classification 2010
Primary: 37P55
Secondary: 37F10
Received: 30 November 2012
Revised: 25 May 2013
Accepted: 4 July 2013
Published: 31 May 2014
Najmuddin Fakhruddin
School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba
Mumbai 400005