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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

具有发散 Birkhoﬀ 正规型的实解析复保面积

变换

尹万科

武汉大学数学与统计学院，湖北武汉 430072

摘要：本文主要证明存在实解析的复保面积变换，在椭圆点附近，有发散的 Birkhoﬀ 正规型。

我们的证明方法主要是小除子理论和 Gong 研究 Halmitonian 系统里的方法。

关键词：多复变函数论，正规型，保面积变换

中图分类号： 0174.56

Divergent Birkhoﬀ normal forms of real analytic

complex area preserving maps

YIN Wan-Ke

School of Mathematics, Wuhan University, Hubei 430072, China.

Abstract: In this paper, we provide a real analytic and complex valued area preserving map,

that possesses a divergent Birkhoﬀ normal form near an elliptic ﬁxed point. The method uses

the small divisor theory and the work of Gong in the study of the Halmitonian system.

Key words: Several complex variables, normal form, area-preserving maps

0 Introduction

We consider a real analytic area preserving map near (0, 0) of the form:

= f(x, y ) = x cos γ

− y sin γ

+ p(x, y),

= g(x, y) = x sin γ

+ y cos γ

+ q(x, y), f

− f

= 1,

(0.1)

where f(x, y) and g(x, y) are real-valued analytic functions in a neighborhood of (0, 0) in R

We also assume that γ

/π is an irrational real number. Then (0, 0) is an elliptic ﬁxed point in

terms of [1]. G. D. Birkhoﬀ [2] (also see §23 of [1] for more details) showed that there is a new

change of coordinates by formal power series, in which the mapping has the following formal

normal form

= ζ cos w − η sin w, η

= ζ sin w + η cos w, w =

+∞



i=0

(ζ

+ η

)

, (0.2)

基金项目： The author is supported in part by NSFC-10901123, SRFDP-20090141120010, FANEDD-201117,ANR-09-

BLAN-0422 and NSFC-11271291.

作者简介： YIN Wan-Ke （1980-），male，associate professor，major research direction：Several Complex Variables and

Complex Geometry.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

In [3] , H. Rüssmann showed that the formal transformation is divergent in general. However,

it is not clear if there is a real analytic area preserving map with a divergent Birkhoﬀ normal

form in the literature.

In this paper, we will consider the following real analytic and complex valued area pre-

serving maps from (x, y) ∈ C

to (x

, y

) ∈ C

= P (x, y) = λx + p(x, y), y

= Q(x, y) = λy + q(x, y), P

− P

= 1. (0.3)

Following the original idea of Gong in [4], which was used by him to construct Hamiltonian

functions with divergent Birkhoﬀ normal forms, we prove, in this paper, the following:

Theorem 1. There exists a real analytic and complex valued area preserving map near an

elliptic ﬁxed point that possesses a divergent Birkhoﬀ normal form.

We would like to mention that Theorem 1 does not follow directly from the divergence

of the normal forms for the Hamiltoninan systems [4], since, for the Hamiltonian systems, the

dimension must be at least 4.

Besides the work of Gong in [4], systematic investigations on the normal form of the

Hamiltonian functions can be found in [5], [6], [7], [8], [9], [10] and the references therein.

However, it is not known how to get the counterpart of these results for the area preserving

mappings. In P. Marco [10], it is proved that if there is a Hamiltonian function with a divergent

Birkhoﬀ normal form, then a generic Hamiltonian function with the same quadratic term also

has a divergent normal form. It is interesting to know whether there is a similar theorem for

real analytic area preserving maps. More precisley, if there is a real analytic area preserving

map with a divergent Birkhoﬀ normal form, is it true that a generic area preserving map with

the same linear term also has a divergent Birkhoﬀ normal formal?

For other related convergence problems encountered in the normal form theory, we refer

the reader to [11], [12], [13] and [14], as well as, the references therein.

1 Preliminaries

We ﬁrst consider a real analytic area preserving map given by the generation function S:

= λx − S

[2]

(λx, y

), y

= λy + S

[1]

(λx, y

). (1.1)

Here λ ∈ C with |λ| = 1 is not a root of unity, S(ς, τ ) is a (complex valued) holomorphic function

of (ς, τ) in a neighborhood of (0, 0) in C

with vanishing order at least 3. Here and in what

follows, we set χ

[1]

(ς, τ ) =

∂χ

∂ς

and χ

[2]

(ς, τ ) =

∂χ

∂τ

for χ being a formal power series. Following

the notations in §23 of [1], we express the transformation (1.1) symbolically as z

= T (z).

- 2 -

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