Periodicities in linear fractional recurrences: Degree growth of birational surface maps

Bedford, Eric; Kim, Kyoung-Hee

doi:10.1307/mmj/1163789919

Cited by 80 publications

(201 citation statements)

References 19 publications

(7 reference statements)

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“…For two distinct such F , say F t , F t ′ , the intersection F t ∩ F t ′ includes C D and hence equals C D by comparing the degree. This proves (1).…”

Section: Proof Of Theorems When P =supporting

confidence: 53%

Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces

Perroni

Zhang

2013

Math. Ann.

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Abstract. We use a concise method to construct pseudo-automorphisms f n of the first dynamical degree d 1 (f n ) > 1 on the blowups of the projective n-space for all n ≥ 2 and more generally on the blowups of products of projective spaces. These f n , for n = 3 have positive entropy, and for n ≥ 4 seem to be the first examples of pseudo-automorphisms with d 1 (f n ) > 1 (and of non-product type) on rational varieties of higher dimensions.

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“…For two distinct such F , say F t , F t ′ , the intersection F t ∩ F t ′ includes C D and hence equals C D by comparing the degree. This proves (1).…”

Section: Proof Of Theorems When P =supporting

confidence: 53%

Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces

Perroni

Zhang

2013

Math. Ann.

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“…by using the property that the globally periodic maps have zero algebraic entropy, it has been proved in [6,7] [42,43]. Again new cases, non-equivalent to the ones of the list given in (6), appear.…”

Section: Theorem 1 ([14])mentioning

confidence: 94%

Different Approaches to the Global Periodicity Problem

Cima

Gasull

Mañosa

et al. 2016

Springer Proceedings in Mathematics &Amp; Statistics

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Let F be a real or complex n-dimensional map. It is said that F is globally periodic if there exists some p ∈ N + such that F p (x) = x for all x, whereThe minimal p satisfying this property is called the period of F. Given a m-dimensional parametric family of maps, say F λ , a problem of current interest is to determine all the values of λ such that F λ is globally periodic, together with their corresponding periods. The aim of this paper is to show some techniques that we use to face this question, as well as some recent results that we have obtained. We will focus on proving the equivalence of the problem with the complete integrability of the dynamical system induced by the map F, and related issues; on the use of the local linearization given by the Bochner Theorem; and on the use the Normal Form theory. We also present some open questions in this setting.

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“…Remarkably, F also leaves invariant a unique irreducible quartic surface Y ⊂ P 3 . This invariant quartic can be compared to the invariant cubic C ⊂ P 2 that plays an important role in the study of dynamics on rational surfaces [BK1], [Mc3]. Both Y and C are anticanonical divisors in their respective projective spaces.…”

Section: K3 Surfaces In Pmentioning

confidence: 99%

“…The lower bound h(F ) ≥ log λ 10 can also be achieved on rational surfaces [BK1], [Mc3] and on non-projective K3 surfaces [Mc4], but not on Enriques surfaces [Og2], complex tori [Mc4,Thm 1.3], or any other types of surfaces [Ca]. In [Ue], the entropy spectrum for rational surfaces is shown to coincide with the Weyl group spectrum.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of projective K3 surfaces with minimum entropy

McMullen

2015

Invent. math.

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Periodicities in linear fractional recurrences: Degree growth of birational surface maps

Cited by 80 publications

References 19 publications

Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces

Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces

Different Approaches to the Global Periodicity Problem

Automorphisms of projective K3 surfaces with minimum entropy

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