A transcendental dynamical degree

Bell, Jason P.; Diller, Jeffrey; Jönsson, Mattias

doi:10.4310/acta.2020.v225.n2.a1

Cited by 13 publications

(17 citation statements)

References 18 publications

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“…In fact, there is a counterexample to Conjecture 2.2(c) due to [LS20]. Furthermore, the results from [BDJ20,BDJK21] suggest that claim (b) may also be false. Let us list some previous works towards claim (d).…”

Section: Preliminariesmentioning

confidence: 96%

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: 96%

Periodic points and arithmetic degrees of certain rational self-maps

Wang¹

2022

Preprint

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“…Another line of research leading to the study of the same structures is the multi-dimensional complex dynamics. A representative though, of course, non-exhaustive list of relevant references includes [5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,29,30,31,32,36,37,49,58].…”

Section: Introductionmentioning

confidence: 99%

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups I. General theory and 2D examples

Alonso¹,

Suris²,

Wei³

2023

Preprint

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In this paper we address the problem of computing deg(f n ), the degrees of iterates of a birational map f : P N P N . For this goal, we develop a method based on two main ingredients: the factorization of a polynomial under pull-back of f , based on local indices of a polynomial associated to blow-ups used to resolve the contraction of hypersurfaces by f , and the propagation of these indices along orbits of f . For maps admitting algebraically stable modifications f X : X X, where X is a variety obtained from P N by a finite number of blow-ups, this method leads to an algorithm producing a finite system of recurrent equations relating the degrees and indices of iterated pull-backs of linear polynomials. We illustrate the method by three representative two-dimensional examples. It is actually applicable in any dimension, and we will provide a number of three-dimensional examples as a separate companion paper.

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“…The list of examples we obtain is infinite, but not completely explicit, and there remain some very interesting further questions. As in [4], our examples are based on monomial maps, that is, maps ℎ 𝐴 ∶ ℙ 𝑑 ⤏ ℙ 𝑑 whose components ℎ 𝐴,𝑗 = 𝑥 𝑎 𝑗1 1 … 𝑥 𝑎 𝑗𝑑 𝑑 are monomials with exponents specified by the 𝑗th row of a 𝑑 × 𝑑 integer matrix 𝐴. Since we aim to construct birational maps, we will always take 𝐴 ∈ SL 𝑑 (ℤ).…”

Section: Introductionmentioning

confidence: 99%

“…4, we can find a Zariski closed subset 𝑍 𝑛 ⊂ 𝕋 of codimension at least two such that 𝐻 ∩ Ind(g 𝑘 ) ⊂ 𝑍 𝑛 for 1 ⩽ 𝑘 ⩽ 𝑛 and g 𝑘 (𝐻 𝑗 ⧵ 𝑍 𝑛 ) ⊂ 𝕋 𝐸 𝑗 for 0 ⩽ 𝑗 ⩽ 𝑑, 1 ⩽ 𝑘 ⩽ 𝑛. Using 𝑍 𝑘 < 𝑛, where the Zariski closure is taken in 𝕋.…”

mentioning

confidence: 99%