Existence of non-preperiodic algebraic points for a rational self-map of infinite order

Amerik, Ekaterina

doi:10.4310/mrl.2011.v18.n2.a5

Cited by 39 publications

(47 citation statements)

References 2 publications

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“…Theorem 2.1. (Amerik [1]) Let X be a projective variety defined over a field k of characteristic zero and let f : X X be a dominant self-map that has infinite order and is also defined over k. Then there exists a point x ∈ X(k) such that orbit of x under f is infinite.…”

Section: Geometric Resultsmentioning

confidence: 99%

“…The proof of Theorem 1.1 uses work of Amerik [1], which is in turn used work of Hrushovski [16], and the work of the first author and Ghioca and Tucker [2]. In particular, given a field k of characteristic zero and a finitely generated extension K of k, a k-algebra automorphism σ of K corresponds to a birational map φ : X X of a normal projective variety defined over k, whose field of rational functions is K. Amerik shows that if σ has infinite order then there is some point x ∈ X(k) whose orbit under φ is defined at every point and such that the orbit is infinite.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Free algebras and free groups in Ore extensions and free group algebras in division rings

Bell

Gonçalves

2016

Journal of Algebra

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Abstract. Let K be a field of characteristic zero, let σ be an automorphism of K and let δ be a σ-derivation of K. We show that the division ring D = K(x; σ, δ) either has the property that every finitely generated subring satisfies a polynomial identity or D contains a free algebra on two generators over its center. In the case when K is finitely generated over k we then see that for σ a k-algebra automorphism of K and δ a k-linear derivation of K, K(x; σ) having a free subalgebra on two generators is equivalent to σ having infinite order, and K(x; δ) having a free subalgebra is equivalent to δ being nonzero. As an application, we show that if D is a division ring with center k of characteristic zero and D * contains a solvable subgroup that is not locally abelian-by-finite, then D contains a free k-algebra on two generators. Moreover, if we assume that k is uncountable, without any restrictions on the characteristic of k, then D contains the k-group algebra of the free group of rank two.

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Section: Geometric Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Free algebras and free groups in Ore extensions and free group algebras in division rings

Bell

Gonçalves

2016

Journal of Algebra

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“…We expect that our techniques may be extended to study other dynamical questions of algebraic nature. The readers can find in a list of open problems, related results and references.…”

Section: Introductionmentioning

confidence: 99%

Growth of the number of periodic points for meromorphic maps

Dinh

Nguyên

Truong

2017

Bull. London Math. Soc.

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We show that any dominant meromorphic self-map f of a compact Kaehler manifold X is an Artin-Mazur map. More precisely, if P_n(f) is the number of its isolated periodic points of period n (counted with multiplicity), then P_n(f) grows at most exponentially fast with respect to n and the exponential rate is at most equal to the algebraic entropy of f. Further estimates are given when X is a surface. Among the techniques introduced in this paper, the h-dimension of the density between two arbitrary positive closed currents on a compact Kaehler surface is obtained.Comment: 18 page

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“…Also, in [36], Zhang formulates a conjecture about the Zariski density of orbits of points under fairly general maps from a projective variety to itself. Amerik, Bogomolov, and Rovinsky [1,2] have obtained partial results towards this conjecture, using p-adic methods similar to those used in this paper. This latter conjecture of Zhang takes the following form in the case of coordinatewise polynomial actions on A g .…”

Section: Conjecture 11 (The Cyclic Case Of the Dynamical Mordell-lanmentioning

confidence: 99%

A case of the dynamical Mordell–Lang conjecture

et al. 2010

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We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. Specifically, let ϕ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of ϕ on (P 1 ) g . If the coefficients of ϕ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (P 1 ) g has only finite intersection with any curve contained in (P 1 ) g . We also show that our result holds for indecomposable polynomials ϕ with coefficients in C. Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over C uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (ϕ, ϕ) on A 2 .

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Existence of non-preperiodic algebraic points for a rational self-map of infinite order

Cited by 39 publications

References 2 publications

Free algebras and free groups in Ore extensions and free group algebras in division rings

Free algebras and free groups in Ore extensions and free group algebras in division rings

Growth of the number of periodic points for meromorphic maps

A case of the dynamical Mordell–Lang conjecture

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