The algebraic dynamics of generic endomorphisms of ℙ<sup><i>n</i></sup>

Fakhruddin, Najmuddin

doi:10.2140/ant.2014.8.587

Cited by 25 publications

(15 citation statements)

References 9 publications

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“…, ϕ m (x m )) for some rational maps ϕ i . In particular, very little is known for arbitrary endomorphisms of quasi-projective varieties, besides the result of Fakhruddin [9], who proved that the Dynamical Mordell-Lang Conjecture holds for generic endomorphisms of P n . In this paper we obtain a very general result for Noetherian spaces in support of Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 99%

The dynamical Mordell-Lang problem for Noetherian spaces

Bell¹,

Ghioca²,

Tucker³

2015

Funct. Approx. Comment. Math.

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Let X be a Noetherian space, let Φ : X −→ X be a continuous function, let Y ⊆ X be a closed set, and let x ∈ X. We show that the set S := {n ∈ N : Φ n (x) ∈ Y } is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety X, any rational map Φ : X −→ X, any subvariety Y ⊆ X, and any point x ∈ X whose orbit under Φ is in the domain of definition for Φ, the set S is a finite union of arithmetic progressions together with a set of Banach density zero. This answers a question posed by Laurent Denis [7]. We prove a similar result for the backward orbit of a point and provide some quantitative bounds.

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Section: Introductionmentioning

confidence: 99%

The dynamical Mordell-Lang problem for Noetherian spaces

Bell¹,

Ghioca²,

Tucker³

2015

Funct. Approx. Comment. Math.

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“…The set of such a ∈ C is contained in the degree-k "Multibrot set" M k , which is a compact subset of C, hence one can further specialize f t to any a ∈ C \ M k . See also [3]-especially [3, §3]-for related results.…”

Section: The First Congruence Holds Because [2] ≡ [5] [Mod 3]mentioning

confidence: 99%

Dynatomic polynomials, necklace operators, and universal relations for dynamical units

Doyle¹,

Fili²,

Hyde³

2021

Preprint

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“…In [2] Amerik proved that dominant endomorphisms which are not of finite order have points of infinite order. The methods of Amerik are inspired by the work of many authors on dynamical systems of varieties over number fields; see for instance [9,11,34,41,78,80]. We will require a mild generalization of Amerik's theorem in which we allow the base field to be an algebraically closed field of characteristic zero (which is not necessarily Q).…”

Section: Endomorphisms Of Arithmetically Hyperbolic Varietiesmentioning

confidence: 99%

Arithmetic hyperbolicity: automorphisms and persistence

Javanpeykar¹

2018

Preprint

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We verify some "arithmetic" predictions made by conjectures of Campana, Hassett-Tschinkel, Green-Griffiths, Lang, and Vojta. Firstly, we prove that every dominant endomorphism of an arithmetically hyperbolic variety over an algebraically closed field of characteristic zero is in fact an automorphism of finite order, and that the automorphism group of an arithmetically hyperbolic variety is a locally finite group. To prove these two statements we use (a mild generalization of) a theorem of Amerik on dynamical systems which in turn builds on work of Bell-Ghioca-Tucker, and combine this with a classical result of Bass-Lubotzky. Furthermore, we show that if the automorphism group of a projective variety is torsion, then it is finite. In particular, we obtain that the automorphism group of a projective arithmetically hyperbolic variety is finite, as predicted by Lang's conjectures. Next, we apply this result to verify that projective hyperkähler varieties with Picard rank at least three are not arithmetically hyperbolic. Finally, we show that arithmetic hyperbolicity is a "geometric" notion, as predicted by Green-Griffiths-Lang's conjecture, under suitable assumptions related to Demailly's notion of algebraic hyperbolicity. For instance, if k ⊂ C is an algebraically closed subfield and X is an arithmetically hyperbolic projective variety over k such that X C is Brody hyperbolic, then X remains arithmetically hyperbolic after any algebraically closed field extension of k.

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The algebraic dynamics of generic endomorphisms of ℙⁿ

Cited by 25 publications

References 9 publications

The dynamical Mordell-Lang problem for Noetherian spaces

The dynamical Mordell-Lang problem for Noetherian spaces

Dynatomic polynomials, necklace operators, and universal relations for dynamical units

Arithmetic hyperbolicity: automorphisms and persistence

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The algebraic dynamics of generic endomorphisms of ℙn

Cited by 25 publications

References 9 publications

The dynamical Mordell-Lang problem for Noetherian spaces

The dynamical Mordell-Lang problem for Noetherian spaces

Dynatomic polynomials, necklace operators, and universal relations for dynamical units

Arithmetic hyperbolicity: automorphisms and persistence

Contact Info

Product

Resources

About

The algebraic dynamics of generic endomorphisms of ℙⁿ