The algebraic dynamics of generic endomorphisms of ℙn

Fakhruddin, Najmuddin

doi:10.2140/ant.2014.8.587

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

Najmuddin Fakhruddin

Vol. 8 (2014), No. 3, 587–608

DOI: 10.2140/ant.2014.8.587

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

Mathematical Subject Classification 2010

Primary: 37P55

Secondary: 37F10

Milestones

Received: 30 November 2012

Revised: 25 May 2013

Accepted: 4 July 2013

Published: 31 May 2014

Authors

Najmuddin Fakhruddin
School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road, Colaba Mumbai 400005 India

Vol. 8, No. 3, 2014