The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

Xie, Junyi

doi:10.1112/s0010437x17007187

Cited by 16 publications

(7 citation statements)

References 31 publications

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“…The proof of the Theorem 6.2 relies on the theory of valuative tree introduced by Favre and Jonsson in [8], which was developed by Favre, Jonsson and the author in [9,10,24,25,26]. 6.1.…”

Section: Endomorphisms Of Amentioning

confidence: 99%

Algebraicity criteria, invariant subvarities and transcendence problems from arithmetic dynamics

Xie¹

2022

Preprint

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We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field K, if it contains many K-points, then it is algebraic over K. This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates.Combing this criteria and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamics. We get a characterization of products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic. For this, we studied the invariant subvarieties for product of endomorphisms. In particular, we partially generate Medvedev-Scanlon's classification of invariant subvarieties of split polynomial maps to separable endomorphisms on (P 1 ) N in any characteristic. We also get some hight dimensional partial generalization via introducing of a notion of independence. We then study dominant endomorphisms f on A N over a number field of algebraic degree d ≥ 2. We show that in most of the cases (e.g. when such an endomorphism extends to an endomorphism on P N ), there are many analytic curves centered at infinity which are periodic. We show that for most of them, it is algebraic if and only if it contains one algebraic point. We also study the periodic curves. We show that for most of f , all periodic curves has degree at most 2. When N = 2, we get a more precise classification result. We show that under a condition which is satisfied for a general f , if f has infinitely many periodic curves, then f is homogenous up to a changing of origin.

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“…The proof of the Theorem 6.2 relies on the theory of valuative tree introduced by Favre and Jonsson in [8], which was developed by Favre, Jonsson and the author in [9,10,24,25,26]. 6.1.…”

Section: Endomorphisms Of Amentioning

confidence: 99%

Algebraicity criteria, invariant subvarities and transcendence problems from arithmetic dynamics

Xie¹

2022

Preprint

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“…This is explained in [39, Theorem 6.2], and [75,Theorem 5.6]. In fact, this version of Amerik's theorem has been used throughout the literature; see for instance [7,Lemma 3.3] and [76].…”

Section: Dominant Self-maps Of Pseudo-mordellic Varietiesmentioning

confidence: 99%

Finiteness properties of pseudo-hyperbolic varieties

Javanpeykar

Xie

2019

Preprint

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Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model, but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi-Ochiai's finiteness result for varieties of general type, and thereby generalize Noguchi's theorem (formerly Lang's conjecture). Our proof here relies on a deformation-theoretic result for surjective maps of normal varieties due to Hwang-Kebekus-Peternell. Contents 19 8. Dominant self-maps of pseudo-Mordellic varieties 21 9. Dominant self-maps of pseudo-1-bounded varieties 22 References 24

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“…In case (3), each point in the orbit of [3 : 2 : 1] under φ 6 is (D, S)-integral for S = {∞, 2, 3}, and it is easy to see from the 2-adic and the 3-adic valuations that an algebraic curve can only contain finitely many points in this orbit. For cases (1) and (2), it follows from a recent work of Xie [35] that there exists an algebraic point whose orbit under φ 6 is Zariski-dense (if there exists ℓ ≥ 1 such that the two eigenvalues of the tangent map at a fixed point of φ 6ℓ are multiplicatively independent, we can invoke [2, Corollary 2.7] instead). By enlarging k and S so that all coordinates of this point are S-units and that all the coefficients of F and G lie in O k,S , we see that each point in this orbit is (D, S)-integral.…”

Section: Integral Points In Orbitsmentioning

confidence: 99%

Integral points and orbits of endomorphisms on the projective plane

Levin

Yasufuku

2018

Trans. Amer. Math. Soc.

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We analyze when integral points on the complement of a finite union of curves in P 2 are potentially dense. We divide the analysis of these affine surfaces based on their logarithmic Kodaira dimensionκ. Whenκ = −∞, we completely characterize the potential density of integral points in terms of the number of irreducible components on the surface at infinity and the number of multiple members in a pencil naturally associated to the surface. When integral points are not potentially dense, we show that they lie on finitely many effectively computable curves. Whenκ = 0, we prove that integral points are always potentially dense. The bulk of our analysis concerns the subtle case ofκ = 1. We determine the potential density of integral points in a number of cases and develop tools for studying integral points on surfaces fibered over a curve. Finally, nondensity of integral points in the caseκ = 2 is predicted by the Lang-Vojta conjecture, to which we have nothing new to add.In a related direction, we study integral points in orbits under endomorphisms of P 2 . Assuming the Lang-Vojta conjecture, we prove that an orbit under an endomorphism φ of P 2 can contain a Zariski-dense set of integral points (with respect to some nontrivial effective divisor) only if there is a nontrivial completely invariant proper Zariski-closed set with respect to φ. This may be viewed as a generalization of a result of Silverman on integral points in orbits of rational functions. We provide many specific examples, and end with some open problems.

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The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

Cited by 16 publications

References 31 publications

Algebraicity criteria, invariant subvarities and transcendence problems from arithmetic dynamics

Algebraicity criteria, invariant subvarities and transcendence problems from arithmetic dynamics

Finiteness properties of pseudo-hyperbolic varieties

Integral points and orbits of endomorphisms on the projective plane

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