Periodic points for area-preserving birational maps of surfaces

Iwasaki, Katsunori; Uehara, Takato

doi:10.1007/s00209-009-0570-3

Cited by 17 publications

(37 citation statements)

References 23 publications

(41 reference statements)

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“…The reader will find some related results in Favre [42], Iwasaki-Uehara [57], Saito [69], Xie [80] and the references therein. Note that in some references, periodic points which are indeterminacy points may not be counted.…”

Section: Theorem 54 (Dinh-nguyen-truong) Let F Be a Dominant Meromomentioning

confidence: 99%

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 54 (Dinh-nguyen-truong) Let F Be a Dominant Meromomentioning

confidence: 99%

Equidistribution problems in complex dynamics of higher dimension

Dinh

Sibony

2017

Int. J. Math.

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“…So if

x

is a periodic point of period

n

f

, then either

f^{n}

is holomorphic near

x

and fixes the point

x

or the similar property holds for

f^{- n}

. Saito's local index function is the function

ν_{•} false(f^{n} false) : {prefixPer}_{n} false(f false) \to N

defined in [, Definition 3.5]. This function is 0 except at a finite number of points.…”

Section: Bi‐meromorphic Maps On Compact Kähler Surfacesmentioning

confidence: 99%

“…In other words,

scriptI

is of complete intersection. By [, Definition 3.5],

ν_{x} false(f false) : = {prefixdim}_{double-struckC} {scriptO}_{0} / I .

The following lemma shows that Saito's index extends the notion of multiplicity for isolated periodic points. Lemma If

x

is an isolated fixed point of

f

as above, then

ν_{x} false(f false)

is the multiplicity of the intersection

Γ (f) \cap Δ

at the point

(x, x)

.…”

Section: Bi‐meromorphic Maps On Compact Kähler Surfacesmentioning

confidence: 99%

“…Despite deep knowledge about complex surfaces, the above basic question is still open. We refer to for some results related to this question. When the topological degree of

f

is dominant, that is,

d_{2} false(f false) > d_{1} false(f false)

, the affirmative answer was recently obtained by the authors in .…”

Section: Introductionmentioning

confidence: 99%

“…Our last main result deals with algebraically stable bi‐meromorphic surface maps

f : X \to X

. To this end, we will make use of the Saito's local index function

ν_{•} false(f^{n} false) : {prefixPer}_{n} false(f false) \to N

, introduced by Saito and Iwasaki–Uehara , which extends the usual local index function on isolated periodic points to all periodic points, see Section 4 below for more details. Define

0true P_{n}^{'} (f) : = \sum_{x \in {Per}_{n} (f)} ν_{x} (f^{n}) .

Clearly,

P_{n} false(f false) ⩽ P_{n}^{'} false(f false)

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Hypergeometric groups and dynamics on K3 surfaces

Iwasaki

Takada

2022

Math. Z.

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Periodic points for area-preserving birational maps of surfaces

Cited by 17 publications

References 23 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

Hypergeometric groups and dynamics on K3 surfaces

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