On the Dynamical Degrees of Meromorphic Maps Preserving a Fibration

Dinh, Tien-Cuong; Nguyên, Viêt-Anh; Truong, Tuyen Trung

doi:10.1142/s0219199712500423

Cited by 35 publications

(42 citation statements)

References 19 publications

(23 reference statements)

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“…The proof of the above theorem in the general case uses in an essential way a computation with positive closed currents and Theorem 3.1 plays a crucial role. We refer to [22,26,30,74,75] for details and some extensions of this result. We also obtained in these works the following result, which has been obtained by Gromov for holomorphic maps [52].…”

Section: Theorem 42 (Dinh-sibonymentioning

confidence: 94%

“…For example, if T is smooth in some open set U, then T ± are also smooth there and the approximation of T ± by smooth positive closed forms on X is uniform on compact subsets of U. Specific needs can be obtained by going through the details of the proof of the above theorem, see [13,22,26]. We will discuss now the notion of super-potentials.…”

Section: Positive Closed Currents Super-potentials and Densitiesmentioning

confidence: 99%

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Equidistribution problems in complex dynamics of higher dimension

Dinh

Sibony

2017

Int. J. Math.

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ABSTRACT. Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree. The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. In this paper, we will survey some results and tools developed during the last two decades. Related concepts, new techniques and open problems will be presented.Classification AMS 2010: 32H, 32U, 37F.

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Section: Theorem 42 (Dinh-sibonymentioning

confidence: 94%

Section: Positive Closed Currents Super-potentials and Densitiesmentioning

confidence: 99%

Equidistribution problems in complex dynamics of higher dimension

Dinh

Sibony

2017

Int. J. Math.

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“…It is equal to the number of points in a generic fiber of

f

. The algebraic entropy of

f

is defined by

h_{a} false(f false) : = \underset{0 ⩽ p ⩽ k}{trueprefixmax} log d_{p} false(f false) .

The topological entropy of

f

is always bounded above by the algebraic entropy, see for details.…”

Section: Introductionmentioning

confidence: 99%

“…The topological entropy of f is always bounded above by the algebraic entropy, see [13,15,16,27,42] for details.…”

Section: Introductionmentioning

confidence: 99%

Growth of the number of periodic points for meromorphic maps

Dinh

Nguyên

Truong

2017

Bull. London Math. Soc.

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We show that any dominant meromorphic self-map f of a compact Kaehler manifold X is an Artin-Mazur map. More precisely, if P_n(f) is the number of its isolated periodic points of period n (counted with multiplicity), then P_n(f) grows at most exponentially fast with respect to n and the exponential rate is at most equal to the algebraic entropy of f. Further estimates are given when X is a surface. Among the techniques introduced in this paper, the h-dimension of the density between two arbitrary positive closed currents on a compact Kaehler surface is obtained.Comment: 18 page

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“…Primitivity of f can also be detected from dynamical degrees via the following criterion (see [36]), which is a consequence of results in [19] and [20]: If λ 1 (f ) = λ 2 (f ) then f is primitive.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of blowups of threefolds being Fano or having Picard number 1

Truong¹

2016

Ergod. Th. Dynam. Sys.

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Abstract. Let X 0 be a smooth projective threefold which is Fano or which has Picard number 1. Let π : X → X 0 be a finite composition of blowups along smooth centers. We show that for "almost all" of such X, if f ∈ Aut(X) then its first and second dynamical degrees are the same. We also construct many examples of finite blowups X → X 0 , on which any automorphism is of zero entropy.The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology classes.We will also discuss a possible application of these results to a threefold constructed by Kenji Ueno.

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On the Dynamical Degrees of Meromorphic Maps Preserving a Fibration

Cited by 35 publications

References 19 publications

Equidistribution problems in complex dynamics of higher dimension

Equidistribution problems in complex dynamics of higher dimension

Growth of the number of periodic points for meromorphic maps

Automorphisms of blowups of threefolds being Fano or having Picard number 1

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