On Degree Growth and Stabilization of Three Dimensional Monomial Maps

Lin, Jan-Li

doi:10.48550/arxiv.1204.6258

Cited by 5 publications

(3 citation statements)

References 2 publications

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“…As shown by Favre [Fa] this is not always possible to achieve. However, for large classes of surface maps and monomial maps, one can find models X ′ → X (with at worst quotient singularities), so that f lifts to a 1-stable map, see [DF,Fa,FJ,L1,JW,L3].…”

Section: Introductionmentioning

confidence: 99%

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Stabilization of monomial maps in higher codimension

Lin¹,

Wulcan²

2014

Ann. inst. Fourier

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A monomial self-map f on a complex toric variety is said to be k-stable if the action induced on the 2k-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of f , we can find a toric model with at worst quotient singularities where f is k-stable. If f is replaced by an iterate one can find a k-stable model as soon as the dynamical degrees λ k of f satisfy λ 2 k > λ k−1 λ k+1 . On the other hand, we give examples of monomial maps f , where this condition is not satisfied and where the degree sequences deg k (f n ) do not satisfy any linear recurrence. It follows that such an f is not k-stable on any toric model with at worst quotient singularities.

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Section: Introductionmentioning

confidence: 99%

“…For m = 2, Theorems A and B follow from [Fa] and for m = 3 they follow from [L3,Theorem 1.1]. Moreover, for k = 1 Theorem A follows from [JW, Theorem A].…”

Section: Introductionmentioning

confidence: 99%

Stabilization of monomial maps in higher codimension

Lin¹,

Wulcan²

2014

Ann. inst. Fourier

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“…In general, we will show by example (see Example 7.11) that it is not always possible to find a modification π : X → (C 2 , 0) for which functoriality (f n ) * = (f * ) n holds for all n ≥ 1. Finding such an X is a local analogue of finding an algebraically stable model for a global complex dynamical system, a very active area of research, see, for instance, [DF01], [Fav03], [BD05], [DS05], [DS09], [FJ11], [JW11], [Lin12a], and [Lin12b]. In this light, Theorem B should be viewed as a guarantee that models always exist on which f satisfies a weak local algebraic stability condition.…”

Section: Introductionmentioning

confidence: 99%

Growth of attraction rates for iterates of a superattracting germ in dimension two

Gignac¹,

Ruggiero²

2014

Indiana Univ. Math. J.

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We study the sequence of attraction rates of iterates of a dominant superattracting holomorphic fixed point germ f : (C 2 , 0) → (C 2 , 0). By using valuative techniques similar to those developed by Favre-Jonsson, we show that this sequence eventually satisfies an integral linear recursion relation, which, up to replacing f by an iterate, can be taken to have order at most two. In addition, when the germ f is finite, we show the existence of a bimeromorphic model of (C 2 , 0) where f satisfies a weak local algebraic stability condition.

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Automorphisms and Dynamics: A List of Open Problems

Cantat

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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On Degree Growth and Stabilization of Three Dimensional Monomial Maps

Cited by 5 publications

References 2 publications

Stabilization of monomial maps in higher codimension

Stabilization of monomial maps in higher codimension

Growth of attraction rates for iterates of a superattracting germ in dimension two

Automorphisms and Dynamics: A List of Open Problems

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