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

Perroni, Fabio; Zhang, De‐Qi

doi:10.1007/s00208-013-0992-4

Cited by 9 publications

(10 citation statements)

References 13 publications

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“…The article [22] considers the more general problem of existence of pseudoautomorphisms F X : X X, where π X : X → (P k ) m is a modification of the multiprojective space (P k ) m , obtained as before by blowing up distinct smooth points along an 'elliptic normal curve' C. Our method yields formulas for the pseudoautomorphisms that arise in this case, too. Hence we conclude with a quick sketch of the computations that arise here, laying greatest emphasis on the way things differ from the work presented above.…”

Section: Pseudoautomorphisms On Multiprojective Spacesmentioning

confidence: 99%

“…Following McMullen's approach, Perroni and Zhang [22] recently showed that one can also construct pseudoautomorphisms F X : X X with δ(F X ) > 1 on point blowups X of P k (and more generally, on point blowups of products P k × · · · × P k ). As in McMullen, they begin with a candidate for the pullback action F * X : Pic (P k ) → Pic (P k ), chosen from a certain reflection group.…”

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confidence: 99%

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Pseudoautomorphisms with Invariant Curves

Bedford

Diller

Kim³

2015

Complex Geometry and Dynamics

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Nontrivial automorphisms of complex compact manifolds are typically rare and more typically non-existent. It is interesting to understand which manifolds admit automorphisms, how plentiful they are on any given manifold, and what further special properties distinguish a particular automorphism, or family of automorphisms. These problems have enjoyed much attention in the past fifteen years, motivated largely by work in complex dynamics (e.g. Cantat's thesis [7]). In this introduction, we give a quick account of some of this research, introducing in particular the more general category of pseudoautomorphisms, which occur more frequently in higher dimensions than automorphisms. The final aim of our paper is to present a concrete alternative approach to some recent existence results [22, Theorems 1.1 and 3.1] of Perroni and Zhang for pseudoautomorphisms with invariant elliptic curves on rational complex manifolds. Our methods lead to explicit formulas which are especially simple (see Theorems 4.7 and 6.4) when the pseudoautomorphisms correspond to the 'Coxeter element' in an infinite, finitely generated reflection group.The topological entropy of an automorphism is a non-negative number that measures the complexity of point orbits. 'Positive entropy' will serve as a precise and reasonable necessary condition for a map to be dynamically interesting. In complex dimension one, i.e. on closed Riemann surfaces, there are no automorphisms of positive entropy. In dimension two, Cantat [7] showed that only three types of complex surfaces can carry automorphisms of positive entropy: tori, K3 surfaces (or certain quotients), or rational surfaces. Automorphisms of tori are essentially linear. The cases of K3 and rational surfaces are much more interesting. Dynamics of automorphisms of K3 surfaces were studied in detail by Cantat [8]. McMullen [17] constructed examples which exhibit rotation domains (two dimensional 'Siegel disks'). The family of all K3 surfaces has dimension 20, and the maximum dimension of a continuous family of K3 surface automorphisms is even smaller. By contrast, there are continuous families of rational surface automorphisms which have arbitrarily large dimension [5].It is known [20,12] that rational complex surfaces X that carry automorphisms of positive entropy are in fact modifications (i.e. compositions of point blowups) π : X → P 2 of the complex projective plane P 2 . Thus a rational surface automorphism F X : X → X with positive entropy descends via π to a birational 'map' F : P 2 P 2 which is locally biholomorphic at generic points but also has a finite union of exceptional curves that are contracted to points and conversely a finite collection I(F ) of indeterminate points which are (in a precise sense) each mapped to an algebraic curve. Since the group of all birational maps F : P 2 P 2 is quite large, this suggests trying to find automorphisms by looking at a promising family of plane birational maps and identifying those elements whose exceptional/indeterminate behavior can be eliminated by ...

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Section: Pseudoautomorphisms On Multiprojective Spacesmentioning

confidence: 99%

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confidence: 99%

Pseudoautomorphisms with Invariant Curves

Bedford

Diller

Kim³

2015

Complex Geometry and Dynamics

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“…Non-cohomologically hyperbolic maps of 3-dimensional manifolds X arise naturally as certain pseudoautomorphisms that are "reversible" on H 1,1 (X) [6,5,32]. For these mappings it follows from Poincaré duality that λ 1 (ϕ) = λ 2 (ϕ).…”

Section: Introductionmentioning

confidence: 99%

Examples of rational maps of $CP^2$ with equal dynamical degrees and no invariant foliation

Kaschner¹,

Pérez²,

Roeder³

2016

Bul. Soc. Math. France

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“…Note that (see Lemma 1) if f is a pseudo-automorphism then so are the iterates f n (n ∈ Z). Recent constructions by Bedford-Kim [9], Perroni-Zhang [42], Oguiso [41] [40], and Blanc [4] provided many interesting examples of such maps. Among bimeromorphic selfmaps, it may be argued that the class of pseudo-automorphisms is the second best after that of automorphisms.…”

Section: Introductionmentioning

confidence: 99%

Some dynamical properties of pseudo-automorphisms in dimension 3

Truong

2014

Trans. Amer. Math. Soc.

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Abstract. Let X be a compact Kähler manifold of dimension 3 and let f : X → X be a pseudo-automorphism. Under the mild condition that λ 1 (f ) 2 > λ 2 (f ), we prove the existence of invariant positive closed (1, 1) and (2, 2) currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equidistribution result (which is essentially known in the literature) for Green (1, 1) currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact Kähler manifold of arbitrary dimension; and discuss how a stronger equidistribution result may be proved for pseudo-automorphisms in dimension 3. As a byproduct, we show that the intersection of some dynamically related currents are well-defined with respect to our definition here, even though not obviously to be seen so using the usual criteria.

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

Cited by 9 publications

References 13 publications

Pseudoautomorphisms with Invariant Curves

Pseudoautomorphisms with Invariant Curves

Examples of rational maps of $CP^2$ with equal dynamical degrees and no invariant foliation

Some dynamical properties of pseudo-automorphisms in dimension 3

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