On the arithmetic dynamics of monomial maps

Lin, Jan-Li

doi:10.1017/etds.2018.5

Cited by 6 publications

(4 citation statements)

References 26 publications

(30 reference statements)

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“…(b)]) for an arbitrary rational map. Known results include regular affine automorphisms ([Kaw06, Kaw13,Lee13], see also [KS14, Theorem 2(b)]) and monomial maps ( [Sil14], see also [KS14,Theorem 2(d)] and [Lin19]).…”

Section: Preliminariesmentioning

confidence: 99%

Periodic points and arithmetic degrees of certain rational self-maps

Wang¹

2022

Preprint

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Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for affine automorphisms. We also study the Kawaguchi-Silverman conjecture concerning the dynamical and the arithmetic degrees for birational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell-Lang conjecture and verify the Kawaguchi-Silverman conjecture for some new cases.

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Section: Preliminariesmentioning

confidence: 99%

Periodic points and arithmetic degrees of certain rational self-maps

Wang¹

2022

Preprint

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“…In general, the set tα f pxq | x P X f pQqu of arithmetic degrees can be infinite (Theorem 2.7), but for surjective self-morphisms on normal projective varieties, it is finite and described by the eigenvalues of f ˚: N 1 pXq ÝÑ N 1 pXq (Theorem 2.9). Lin determined the set of arithmetic degrees of monomial maps on projective toric varieties [34,Theorem A]. In [39], we determined the set of arithmetic degrees of surjective self-morphisms on semi-abelian varieties.…”

Section: Related Topicsmentioning

confidence: 99%

“…For surjective self-morphisms on projective varieties, this sequence is well-understood, at least when α f pxq ą 1 (See [54,Theorem 1.1]). For rational maps, this sequence behaves wildly (See [34,Example 3.1]), making it intriguing to discern some patterns.…”

Section: Future Directionsmentioning

confidence: 99%

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties

Matsuzawa

2020

Advances in Mathematics

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We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group endomorphisms of linear algebraic groups. We also propose a possible approach to the conjecture using equivariant MMP.Cartier divisors D). By definition, N 1 (X) and N 1 (X) are dual to each other.-When X is normal, the Iitaka dimension of a Q-Cartier divisor D on X is denoted by κ(D). • Let M be a Z-module. We write M Q = M⊗ Z Q, M R = M⊗ Z R, and so on.

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“…In [14,Theorem 1.6], Shibata and we proved that for any surjective morphism f , there exists a point x ∈ X such that α f (x) = δ f . When X is a toric variety and f is a self-rational map on X that is induced by a group homomorphism of the algebraic torus, the set A(f ) is completely determined [18,12]. When X is quasi-projective, the arithmetic degrees and dynamical degrees can be defined by taking a smooth compactification of X.…”

Section: Introductionmentioning

confidence: 99%