Algebra & Number Theory

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

Najmuddin Fakhruddin

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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.

Article information

Algebra Number Theory, Volume 8, Number 3 (2014), 587-608.

Received: 30 November 2012
Revised: 25 May 2013
Accepted: 4 July 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37P55: Arithmetic dynamics on general algebraic varieties
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]

generic endomorphisms projective space


Fakhruddin, Najmuddin. The algebraic dynamics of generic endomorphisms of $\mathbb{P}^n$. Algebra Number Theory 8 (2014), no. 3, 587--608. doi:10.2140/ant.2014.8.587.

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