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THE ORBIT STRUCTURE OF PSEUDOGROUP ON RIEMANN SURFACES VS. DYNAMICS OF ALGEBRAIC CURVES(Singularities of Holomorphic Vector Fields and Related Topics)

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THE ORBIT STRUCTURE OF PSEUDOGROUP ON RIEMANN SURFACES VS. DYNAMICS OF ALGEBRAIC CURVES

ISAO NAKAI

Department of Mathematics

Hokkaido University

ABSTRACT. Weinvestigate the dynamical propertiesofalgebraic plane curvesin $\mathbb{P}\cross \mathbb{P}$, and

discuss the relation with psudogroup of diffeomorphisms of thecurvesgenerated by the

mon-odromy actions.

INTRODUCTION

Let $\Gamma$ be a pseudogroup consisting of diffeomorphisms

$f$ : $U_{f},$$0arrow f(U_{f}),$$0$ of open

neighbourhoods $U_{f}$ of the complex plane $\mathbb{C}$ which fix $0\in \mathbb{C}$

.

We call the

group

$\Gamma_{0}$ of the

germs

of those $f\in\Gamma$ the germ of $\Gamma$, and call $\Gamma$ a representative of $\Gamma_{0}$

.

$\Gamma$ is non-solvable ifits

germ

is non-solvable.

We say that a subset $A\subset \mathbb{C}$ is invariant under $\Gamma$ if $f(A\cap U_{f})=A\cap f(U_{f})$ for all

$f\in\Gamma$

.

We call a minimal invariant set an orbit (which is not necessarily closed). The

orbit containing an $x$ is unique and denoted $\mathcal{O}(x)$, which is the set of those $f(z)$ with

$z\in U_{f},$$f\in\Gamma$

.

Let $B_{f}$ denote the set of those $z\in U_{f}$ such that $f^{(n)}(z)arrow 0$ as $narrow\infty$

,

where $f^{(n)}$ stands for the n-iterated $fo\cdots of.$ If $f$ has the indifferent linear term $z$ (in

other words parabolic or

flat

at the origin), $B_{f}\cup B_{f^{(-1)}}$ is

an

open neighbourhood

of

$0$

(Proposition 2.4). The basin $B_{\Gamma}$ is the set of points $z$ for which the closure of the orbit

contains the origin. Proposition

2.5

asserts that thebasin is an open neighbourhood ofthe

origin if an $f\in\Gamma$ is flat.

Assume $B_{\Gamma}$ is an open neighbourhood of $0$

.

The separatrix $\Sigma(\Gamma)$ for $\Gamma$ is aclosed real

semianalytic subset of$B_{\Gamma}$, which possesses the following properties.

(1) $\Sigma(\Gamma)$ is invariant under $\Gamma$ and smooth off $0$,

(2) The germ of $\Sigma(\Gamma)$ at $0$ is holomorphically diffeomorphic to a union of $0$ and some

branches ofthe real analytic curve ${\rm Im} z^{k}=0$ for some $k$,

(3) Any orbit is dense or empty in each connected component of$B_{\Gamma}-\Sigma(\Gamma)$

,

(4) Any orbit is dense or empty in each connected component of $\Sigma(\Gamma)-0$

.

Key words andphrases. Pseudogroup, group action, orbit, holomorphic diffeomorphism.

(2)

Corollary 2. If the

germ

$\Gamma_{0}$ is non-solvable and the subgroup $\Gamma_{0}^{0}$ of the germs of

dif-feomorphisms $h\in\Gamma_{0}$ with the indifferent linear term does not admit antiholomorphi$c$

involution, then $\Sigma(\Gamma)-0$ and all orbi$ts$ different from $0\in \mathbb{C}$ are dense or empty on a

neighbourhood of$0$

.

Example 3. Let

$f(z)=z/1-z,$

$U_{f}=\mathbb{C}-\{1<{\rm Re} z, {\rm Im} z=0\},g(z)=\log(1+z)=$

$z-1/2z^{2}+1/3z^{3}-\cdots$ and let $U_{g}(\ni 1)$ be a smallneighbourhoodof$0$, on which$g$ restricts

to a diffeomorphism onto the image $g(U_{g})$

.

Let $\Gamma$ be the pseudogroup generated by $f$ and

$g$

.

Since on the z-plane, $\tilde{z}=1/z,$ $f$ induces the translation by $-1$, the basin $B_{\Gamma}$ is the whole plane C. By Theorem 1.8, the

group

$\Gamma_{0}$ generated by the germs of $f,$

$g$ at $0$ is non

solvable. The real lime $\mathbb{R}$ is invariant under the

group

$\Gamma_{0}$ and there is no other invariant

curves. If$U_{g}$ is small enough: $U_{g}\cap \mathbb{R}$ is contained in the halfline $\{-1<{\rm Re} z, {\rm Im} x=0\}$,

then $g$ maps the real line $U_{g}\cap \mathbb{R}$ into $\mathbb{R}$hence $\mathbb{R}$ is invariant under $\Gamma$

.

Clearly $\Gamma$ preserves

the upper (respectively lower) halfplane. Therefore we obtain

$\Sigma(\Gamma)=\mathbb{R}$

.

Next extend $g$ so that the domain of definition $U_{g}$ intersects with the half line $\mathbb{R}_{-1}^{-}=$

$\{{\rm Re} z<-1, {\rm Im} x=0\}$

.

Then $g$ maps the intersedtion $U_{g}\cap \mathbb{R}_{-1}^{-}$ into the complement

of the real line, where all orbits are locaUy dense. The local density holds also at the

intersedtion $U_{g}\cap \mathbb{R}_{-1}^{-}$ and propergates to the negative part ofthe real line $\mathbb{R}^{-}$ Therefore

we obtain

$\Sigma(\Gamma)=\mathbb{R}^{+}$

.

Let $C\subset \mathbb{C}\cross \mathbb{C}$ be an irreducible algebraic curve and

$\pi_{x},$$\pi_{y}$ be the natural projection

onto thefirst and the second coordinates. Assume that $C$ is nondegenerate i.e. $C$does not

contain a fiber of the projections. The equivalence relation\sim introducedon $C$is generated

by the relations

$p\sim q$ if $\pi_{x}(p)=\pi_{x}(q)$ or $\pi_{y}(p)=\pi_{y}(q)$

.

Our problem is

Problem 4. Study the structure of the equivalence dasses.

In order to observe the structure of the dynamics at infinity, we suppose $C\subset \mathbb{P}\cross \mathbb{P}$

.

Level sets of a rational function on $C$ form a pencil of zero divisors.

So

a morphism of$C$

into$\mathbb{P}\cross \mathbb{P}$ is

determined

by a pair of pencils $(L, M)$ of zero divisors on $C$

.

Our

problem is

to study the geometry ofthe morphisms in terms of divisors in pencils.

In the paper [N1] the author proved

Theorem 5(Rigidity theorem). Let $E,$$F$ be non degenerate linear systems of

(3)

into the dual $sp$aces of$E,$$F$ are not of degree 3. Let $h$ : $Carrow C’$ be an orientation

pre-serving homeonorphism sending $di$visors in $E$ to those in F. Then $h$ is a holomorphic diffeomorphism.

This theorem tells that complex space curves are determined by thestructure of

hyper-plane sections. Our problem can regarded as the spacial case of the above theorem. The

theorem suggests

Question 6. Let $C,$ $C’$ be irreducible and nondegenerate algebraic $cu$rves in IF’ $\cross \mathbb{P}$

.

Let

$\phi,$$\psi$ be orientationpreserving homeomorphisms of$\mathbb{P}$ such that $\phi\cross\psi(C)=C’$

.

Then are

$\phi,$$\psi$ Mobius transformations?

Insome special cases the answer is affirmative. From now onwe seek amethod to study

the structure of the equivalence classes.

If these projections restrict respectively to $d$ and $e$ sheeted branched coverings of $C$

,

we say $C$ is of degree $(d, e)$

.

Two curves $C,$ $C’$ are equivalent if $C’=\phi\cross\psi(C)$ with

M\"obius transformation $\phi,$$\psi$ of the coordinate lines. Let $p$ be a singular point of $C$ in

the sense that either of the coordinate projections of $C$ is not nonsingular and the local

multiplicity of the first and second projections are respectively $d’,$$e’$

.

Let $t$ be a local

coordinate of the curve centered at $p$. Then the monodromy action of the first and second

projections are respectively order $d’,$$e’$ and generate a pseudogroup of diffeomorphisms

of open neighbourhoods of $0$ in the $t$ line. The orbit of

$p$ under this pseudogroup $\Gamma_{p}$ is

contained in the equivalence class $\mathcal{O}(p)$ under the relation above introduced. Ifthe germ

$G_{p}$ of the pseudogroup is non-solvable,the orbits under $\Gamma_{p}$ are dense in some sectors of the

basin $B_{\Gamma_{p}}$ of the pseudogroup by Theorem 1. By Corollary 2 if the

germ

of thecurve $C$ at

$p$ does not admit a

germ

ofantiholomorphic involution oftype $\phi\cross\psi$

,

the orbits under $\Gamma_{p}$

are dense in a neighbourhood of $B_{p}$

.

Define the basin $B_{p}$ of$p$ by the set of those $q$ such

that the topological closure of the equivalence class $\mathcal{O}(q)$ contains

$p$

.

It is easy to see that

if the germ $G_{p}$ is non-commutative, the basins $B_{\Gamma_{p}}\subset B_{p}$ are open. The above density of

orbits propagates to the the basin $B_{p}$

.

If the germ is commutative, it is holomorphically

conjugate with the cyclic subgroup of the linear $S^{1}$ action.

In the paper [N1], the author classified the commutative and nonsolvable

groups

of

germs

ofholomorphic diffeomorphisms of the complex plane at the

origin.

It would be a good exercise to classify all algebraic

curves

singular at the origin $p=0\cross 0$ where the

group

$G_{p}$ is nonsolvable [N2]. A subset $K\subset C$ is invariant if$K$ is a union of equivalence

classes. Clearly the basin is invariant.

proclaimProblem Studythe complement ofthe basinsof the singular points ofalgebraic

curves.

Example 7. Assume $C$ is deffied by $(y-x^{2}-c)(y-x)=0$

,

which is the union of th$e$

diagonal line$y=x$ and the parabola$y=x^{2}+c$. By the equivalencerela$tion\sim the$points $(x, y)$ on the parabola areidentified with those points$(x, x)$ on the diagonalline. Therefore

the first projection of the orbi$ts$ aregenerated by the relations $x\sim x^{2}+c$ and $x\sim-x$

,

which are the $u$nion of the foreword orbit and its backword orbits. It is $d$assically known

that the cluster point set ofany backword orbit is Julia set of the dynamics $xarrow x^{2}+c$

.

In thefollowingwe present some genericorbits of the dynamicson the various algebraic

(4)

.

.

.

.

The projection of an orbit on the curve

$x^{3}+x^{2}+0.1xy+y^{2}+x=0$

(5)

$-’:.r_{\backslash }^{\backslash ;}’$

.

$\cdot$

..

. $\cdot$ . . $\backslash ^{4}i:_{:_{\dot{\sim}_{\sim}}}\cdot..$ . $-*$ ’ $1..$

:.

$\cdot$ $\backslash \backslash ^{\backslash }$

. $\cdot:..=_{\backslash }::.$

.

.

.

$:\cdot:\backslash :.\acute{t*_{i}..\cdot\dot{\searrow}.\backslash }..\cdot:....:_{\dot{j}^{1},:}-.:\backslash :_{\backslash }..\cdot\dot{\sim}_{=^{\backslash :_{:^{i_{\backslash }}}}\dot{\cdot}:^{==\cdot:}}’’\cdot\cdot...:_{\backslash ^{-}:}^{--}:’.\cdot.\cdot.\backslash \cdot..\cdot.\cdot.\cdot\cdot..$ .

, .

.

. .

.

:.

$”.=,.\cdot*...\cdot\backslash .\cdot:.\dot{S}^{*.9_{-}}:_{\vee i=}-..\cdot\ldots\cdot.-\cdot’=\iota:::_{\backslash }^{n_{\backslash }}\prime_{-}\backslash :..\cdot..\cdot\cdot...\cdot.\cdot.\cdot r_{i}\cdot.$

:

$\cdot:\backslash =\backslash .\cdot\cdot\cdot:\backslash :_{:}\cdot.\cdot-\cdot:-.:\cdot;_{\backslash ^{\backslash \cdot\prime\backslash _{*J-:}}}\backslash \cdot\cdot\cdot:\overline{g}\cdot.:^{-i_{-.\wedge}}..\cdot..\cdot$

.

.

:

$–a_{:}.\cdot|\backslash \backslash$

:.

.

.

. $-’.=:..:$

:

.

...

$\backslash \cdot\backslash \backslash ::::$

:

$-$

..

.

$l$

The projection of an orbit on the curve

$(1+2i)x^{3}+(3+2i)x^{2}y+(1+2i)xy^{2}+(3+2i)y^{3}+1=0$

(6)

.

:

The projection of an orbit on the curve

$x^{2}y+y^{3}+1=0$

(7)

The projection of

an

orbit on the curve (Julia set)

$(y-x^{2}+0.7-0.3i)(y-x)=0$

(8)

-. $:\vee$

: .

.

.

$\backslash$

The projection of an orbit on the curve

$(y-x^{2}+0.7-0.3i)(y-x)+O.01=0$

(9)

$\backslash \triangleright$:,

$\neg i’\sim..\backslash \cdot$.

$\bigwedge_{\sim^{\nu\prime}}$: :

.

$i_{\wedge}$; . ., $Y$

;.

.. :

$\prime_{i}^{\wedge\vee^{\backslash }}\backslash l_{?}^{\frac{\dot{l}}{\dot{\triangleleft}^{l}\prime}:_{:}}..:^{\backslash \cdot!’}-\cdot.\cdot\cdot.s\not\in_{\iota_{-\cdot\backslash ^{-}}^{\theta_{\backslash }.\dot{\theta}^{-\backslash ^{\dot{S}_{f_{=}}}},}}^{:_{i_{\backslash \prime}^{\prime,}}}:^{\backslash }\backslash ^{\backslash }.\cdot...\cdot.\cdot.\cdot.\cdot....\cdot:..\cdot\cdot\cdot\cdot.$ $\cdot..\cdot$ $-’-$

.

.

. $:^{\backslash }-$ $’:-:.$. . $=a$: $:^{:^{=}}\cdot c^{\backslash _{c}}$.

..

. $\bigvee_{=}$. . :. $’-$ $\tau_{c}$ .

..

.

: :. : . $\cdot$ , .

:..

-. $\backslash$ $r.$. ’ .

.

. . . $=$

..

$\sim$ $-$.

.

..

’ $\backslash$ $*,$

.

$\backslash \backslash \cdot$

. $arrow$

.

$\backslash$

$-=$.

The projection of an orbit on the curve

$(y-x^{2}-0.3)(y-x)+0.3iy^{2}=0$

(10)

. . . $’\backslash$. 4” $\backslash$ / ’. ’. :

.

$\cdot$ $l^{\searrow}$ $.\nwarrow_{\backslash }:.$ , ’ $($ $i$ .

The projection of an orbit on the curve

$(y-x^{2}-0.3)(y-x)-0.03+0.03i=0$ onto the x-plane.

(11)

The projection ofan orbit on the curve

$(y-x^{2}-0.3)(y-x)-0.03+0.03i=0$

(12)

The projection of an orbit on the curve

$(y-x^{2}-0.3)(y-x)-0.03+0.03i=0$

(13)

The projection of an orbit on the curve

$(y-x^{2}+0.3)(y-x)-0.3=0$ onto the x-plane.

(14)

The projection of an orbit on the curve

$(y-x^{2}+0.3)(y-x)-0.3=0$

(15)

The projection ofan orbit on the curve

$(y-x^{2}+0.3)(y-x)+0.1iy^{2}=0$

(16)

The projection of an orbit on the curve

$(y-x^{2}+0.3)(y-x)+0.1iy^{2}=0$

(17)

:

The projection of an orbit on the curve

$(y-x^{2}+0.3)(y-x)-0.03+0.03i=0$ onto the x-plane.

(18)

The projection of an orbit on the

curve

$(y-x^{2}+0.3)(y-x)-0.03+0.03i=0$

(19)

.

.

.

The projection of an orbit on the curve

$(y-x^{2}+0.3)(y-x)+0.03-0.03i=0$ onto the x-plane.

(20)

.

..

.

$f_{\backslash \wedge}^{-}r^{:^{\prime:^{;_{i^{\backslash }}}..-}}\cdot\dot{\overline{r}}_{-.\cdot i}^{\grave{arrow}_{\backslash ^{\backslash }}...\backslash ..r,\backslash };^{i}..\cdot\cdot..\cdot$

.

$\cdot.\cdot.\cdot’\Im_{:\cdot.\backslash ,}^{:_{\backslash }}\backslash ,\cdot$

$=...:..i$ $\overline{f}’$ $’:^{J}J$ $-\backslash$, $\backslash$

.

$\succ_{\vee}\lrcorner j$

.

.: $\prime^{-}l^{\backslash ^{l^{1’}}}$ ’ / $-\backslash (..$ $:_{j}\backslash$ $-$

.

$\backslash ’.J_{\backslash }\dot{\text{ノ}}$

$:’\backslash \backslash$: ’ $i.:_{:}$

.

$j$ ’ $;^{*}$ ’

.

$\dot{J}^{r}=$

;

$-.\prime J_{*}$ :

:

$\backslash$ :

$f_{r_{-..\cdot..\cdot..\dot{J}^{l}}}’\nwarrow_{:^{*}}:_{L,.:^{J\cdot:_{C^{:,\iota_{l\wedge^{Y^{\vee}}}}}}\cdot\backslash _{.}j^{\iota_{\iota}}\tau.\vee-}-\grave{:}.’\backslash \backslash ^{\backslash ,}\dot{l}\backslash ’$

The projection ofan orbit on the curve

$(y-x^{2}-0.3)(y-x)+0.03-0.03i=0$

(21)

.

.

The projection of an orbit on the

curve

$(y-x^{2}-0.3)(y-x)+0.03-0.03i=0$

(22)

. $-$

$=f_{:}^{-\dot{f}_{=}\prime},:..\dot{\kappa}_{-../^{j\prime_{\backslash }’}}\dot{\sim}\backslash ,\cdot..\cdot\cdot.\cdot\cdot\dot{d}^{:..i_{:}’\backslash .\cdot\overline{:^{\backslash }}.\ddot{\ddot{\ovalbox{\tt\small REJECT}}}_{\backslash }}=,\cdot..\cdot...\cdot..\cdot.\cdot\cdot..\cdot.\cdot.\nwarrow_{\backslash }i_{=^{-}}^{\dot{f}’.\cdot:})_{\dot{P}^{:}..\prime_{\backslash }.\ddot{d}^{=}}\backslash _{\backslash }.\cdot..\cdot=:_{\backslash }.\backslash \wedge..\backslash \sim,$

.

$\cdot.\cdot.\dot{j}\nearrow-’/\cdot\prime i’$

$\backslash ^{\backslash =-}\dot{\backslash }=\mathfrak{k},’\prime’;^{:}$

.

.

$\prime’\grave{\prime}.\nearrow\nearrow^{\backslash }\prime i;_{s}$

:

:.’

:

.”

$”\prime^{\prime^{\sim}}\nwarrow^{:}\backslash _{\grave{\searrow}_{\cap^{\backslash }.a^{J\backslash }J’}^{--}\nearrow_{\backslash }^{:}:}\backslash ...\cdot.\cdot..\cdot.\cdot\cdot.\cdot.\cdot...\cdot.\cdot.\ldots\cdot...\cdot..\cdot.\cdot.\cdot..\cdot..\cdot.\cdot\cdot..\cdot\cdot\zeta_{\backslash }^{-f_{:}’}\underline{a}\backslash ^{\vee}\sim_{:^{\backslash }}^{\backslash }\sim^{\backslash ^{j’}}’-:\backslash \prime^{\dot{P}^{:’}}’:$

.

$.$.

$\sim_{\backslash }$

$-$ .

The projection of an orbit on the curve

$(y-x^{2}+0.3i)(y-x)-0.03+0.03i=0$

(23)

.

The projection of an orbit on the curve

$(y-x^{2}+0.3)(y-x)-0.03+0.03i=0$

(24)

.

.

..

The projectionof an orbit on the curve

$(y-x^{2})(y-x)+0.7x^{2}=0$

onto the x-plane.

REFERENCES

a

I. Nakai, Separatricesfor non solvable dynamics on C.O, Ann. 1‘Inst. Fourier 44 (2).

Z

I. Nakai, Topology of complex webs of codimension one and geometry of projective space curves, Topology 26, 475-504.

3

preparation

$_{\vee}^{I.Nak\epsilon}$ Preprint IHES July, Topology of websofdivisors in a linear system of a complexmanifold..

SAPPORO 060 JAPAN

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