Quasiregular analogues of critically finite rational functions with parabolic orbifold

Q U A S I R E G U L A R A N A L O G U E S O F C R I T I C A L L Y F I N I T E

R A T I O N A L F U N C T I O N S W I T H P A R A B O L I C O R B I F O L D

VOLKER MAYER

A b s t r a c t . We study uniformly quasiregular mappings o f ~ , i.e., quasiregular mappings f with uniform control of the dilatation of all the iterates fk, which are analogues of critically finite rational functions with parabolic orbifold. They form a rich family ofnon-injective uniformly quasiregular mappings. In our main result we characterize them among all uniformly quasiregular mappings as those which have an invariant conformal structure flat at a point of a repelling cycle.

1 I n t r o d u c t i o n

In the field of iteration of rational functions there is a family of rational functions

which behaves particularly nicely: the so-called critically finite rational functions

with parabolic orbifold. Latt~s studied such a function [La], and his example was

the first having the whole Riemann sphere as its Julia set.

Lattes' construction of chaotic rational functions extends to higher dimensions

because of its geometric nature. It leads to uniformly quasiregular mappings o f ~ n

with Julia set the whole n-sphere [My]. Such a mapping is obtained as a solution

of Schr6der's equation f o h = h o A, where h: R '~---, ~-~ is a quasiregular mapping

automorphic with respect to some crystallographic group F C Isom(~ '~) and A

is an appropriate similarity (for example, one might take the choice of Latt6s: = 2 x ) .

In this way, one can get other uniformly quasiregular mappings which are

analogues of critically finite rational functions with parabolic orbifold. In order

to do this, it suffices to choose other automorphic quasiregular mappings h. We

call any mapping f obtained in this manner of Latt~s type. A precise definition

and description of mappings of Latt~s type as well as examples that behave like

Tchebychev polynomials are the content of Section 3.

Our main aim is to characterize Latt~s-type mappings. This can be done in

terms of invariant conformal structures, which we recall briefly here. More details

105 JOURNAL D'ANALYSE MATH]EMATIQUE, VoL 75 (1998)

106 v. MAYER

are given in Section 4.

Let f be a uniformly quasiregular mapping and # an f-invariant conformal

structure. This means that # is a measurable mapping from R-~ into the space S(n) o f n x n real positive definite matrices with determinant 1 which satisfies

= , 2 ? , y, . ~, = f ' [ # o f ] (det f ) -~ ( # o f )

Such a structure always exists [IM]. We will say that # is fiat at a point p, if there

is a quasiconformal mapping ~ defined in a neighborhood W o f p such that

~'(x)[Id] = #(x) for a.e. x E W.

Thus flatness here means always quasiconformal flatness. Note that, by Liouville's

Theorem, an f-invariant conformal structure cannot be globally flat, except when

f is homeomorphic. It turns out that local flatness at a point of a repelling cycle

can occur only for Latt6s-type mappings. Moreover, this characterizes them.

M a i n T h e o r e m A non-injective uniformly quasiregular mapping f of ~'~, n > 3, is of Lattbs type ifand only if it has anf-invariant conformalstructure which is flat at a repellingfixed point of some iterate fk of the mapping.

R e m a r k s This condition about the flatness of # deserves some clarifications.

1. Flatness of the conformal structure at a repelling fixed point implies directly

that there is a local quasiconformal change of coordinates such that f becames

a similarity (see Lemma 6.1). Hinkkanen and Martin [HM] showed that

a quasiconformal linearization is always possible in a neighborhood of a

repelling fixed point.

2. Flatness of the conformal structure is a trivial condition in dimension n = 2

because of the measurable Riemann Mapping Theorem. In particular, any

rational function satisfies this condition, which shows that the above theorem

cannot be valid in C.

3. For injective mappings f , the Main Theorem and also Theorem 1 and The-

orem 3 can be viewed as special cases of the results in [Tu] concerning

quasiconformal conjugation of quasiconformal groups to M6bius groups.

We call the structure # obtained by a push-forward of the standard structure

ld by the associated automorphic mapping h the canonical conformal structure of

a Latt6s-type mapping. One of the steps of the proof of the Main Theorem is to

show that, if p is the canonical conformal structure of a Latt~s-type mapping, then

every quasiregular mapping of ~~ having # as an invariant structure is of Latt~s

QUASIREGULAR ANALOGUES 107

type (Proposition 5.1). This rigidity result is also in contrast with the case of the

space of rational functions.

Our next point is to look for conditions that imply the flatness of an invariant

conformal structure and that therefore force the mapping to be of Latt~s type. In

the context of quasiconformal groups, P. Tukia [Tu] showed that certain regularity

of # yields flatness. His method applies to the non-injective setting and leads in

particular to the following result.

T h e o r e m 1. Let f be a non-injective uniformly quasiregular mapping o f F ,

n > 3 and # an f-invariant conformal structure. Suppose that lz is approximately

continuous at a point o f a repelling cycle o f f ; then f is o f Latt~s type.

Note that it is important here that the conformal structure have some smooth-

ness. In fact, smoothness of the mapping alone is not sufficient. The example of

Iwaniec and Martin [IM] has a repelling fixed point and is smooth there, but it is

not of Lattrs type.

2 Definitions and basic properties of quasiregular mappings

Let D C I~ '~ be a domain and f : D - ~ ~ t '~ a mapping of Sobolev class Wtlo'~(D). We consider only orientation preserving mappings, which means that the Jacobian

determinant Jr(z) >_ 0 for a.e. x E D. Such a mapping is said to beK-quasiregular,

where 1 < K < c~, if

max If'(x)hl < K min If'(x)hl for a.e. x E D. Ihl=l thl =1

The smallest number K for which the above inequality holds is called the linear dilatation of f . A non-constant quasiregular mapping can be redefined on a set

of measure zero so as to make it continuous, open and discrete; and we shall

always assume that this has been done. If D is a domain of the compactification

R-~ (equipped with the spherical metric, so that ~ is isometric via stereographic

projection with the n-sphere S'~), we use the chart at infinity x ~ x/ Ix l 2 to extend

in the obvious manner the notion ofquasiregularity to mappings f : D ~ - ~ . Such

mappings are also said to be quasimeromorphic. A mapping f of a domain D into

itself is called uniformly quasiregular if there exists 1 < K < oo such that all the

iterates fk are K-quasiregular. We abreviate this by f E UQR(D). A central result in the theory ofquasiregular mappings is the following [Ge, Re].

L iouvi l le ' s T h e o r e m . In dimension n > 3, every 1-quasiregular mapping

f : D ~ ' ~ is the restriction ofa Mdbius transformation.

108 ~ MAYER

In particular, every 1-quasiregular mapping is injective and therefore the branch set By, i.e., the set of points z E D for which f is not locally homeomorphic at

z, is empty. For the definition of the degree we refer to [Ri]. For our purpose, it

suffices to know that the degree o f a quasiregular mapping f : R'~--.~~ is precisely

the number ofpreimages of any point y E ~n\f(By); see [Ri].

We often use the fact that the complement of the branch set is a domain. The

following discussion follows once more [Ri, p.18]. I f f : ~ ~ - ' ~ is quasiregular,

then the topological dimension of the branch set satisfies dim By < n - 2. Since

By is closed, f~ = I~ n \ By is a domain. Furthermore, i fg : ~ - ' ~ ' ~ is also quasi-

regular, then dim By = dim f (B l ) = dim g-l(f(By)). This is the content of

[CH, 2.1 Lemma]. Consequently, the complement of each of the sets By, f (Bf ) , g-~ (f(Bf)) is a pathwise connected domain.

We end this section by recalling some basic notions of iteration. The dynamical

behaviour of a uniformly quasiregular mapping of ~'~splits the sphere into two

parts: the Fatou set F(f) , which is the set of points z for which {fk} is a normal

family in a neighborhood of x, and the Julia set d(f) = ~'~ \ F(f) . The Fatou set

is open, the Julia set is closed and they are both completely invariant. Moreover,

if the mapping f has a non-empty branch set, then the Julia set cannot be empty.

See [IM] for this and for a more detailed presentation.

Superattractingfixedpoints are fixed points which are also branch points. From

the local behaviour of quasiregular mappings it follows that a superattracting fixed

point p is attracting, i.e., p has a neighborhood U such that f(U) c U (of. [My]).

In particular, it follows that in a neighborhood of a superattracting fixed point z

the iterates fk converge uniformly to this point. Let f~ = {y E ~ ; fk(y) ~ z} be the basin o f attraction of z and f/; the component of f~x containing z. If z is

completely invariant, so that f ( x ) = f - l ( z ) = {x), then the immediate basin o f attraction f~; coincides with f~.

We call p a repelling fixed point of f if f(p) = p and p has a neighborhood U

such that the restriction o f f to U is injective and satisfies U c f(U). A point p is

a point o f a repelling cycle if it is a repelling fixed point o f some iterate f~.

Finally, recall that the exceptional set Ef is the largest discrete completely invariant subset o f ~ "n.

3 D e s c r i p t i o n o f Lat t~s - type m a p p i n g s and e x a m p l e s

3.1 Automorphic mappings The basic tool in the construction of Latt~s- type mappings are automorphic mappings. We say that a quasiregular mapping h


from I~ n into ~'~ is automorphic with respect to a discrete group F of isometr ies o f

I~ '~, i f

(i) h o "r = h for every "r E 1-', and

(ii) F acts transitively on fibers Oy = h - I (y ) .

So we use here a strong form of automorphy. The second point is added to

the usual definition and means that for every two points xx, z2 with h(:cl) = h(z2)

there is an isometry 7 E F identifying them: x2 = 7(xl) .

To describe the discrete groups that may occur we recall some facts o f

Bieberbach's theory. To a discrete group F o f isometries o f ~n we associate

T, the subgroup o f all the translations o f F , and R = F/T, the group o f the rotation

parts o f the elements o f F. The translation group T is isomorphic to Z k , for some

k E {0, 1, ..., n}, and R is a finite group. See [Rw] for more information.

The following is an immediate consequence o f a result of Martio [Mo]. In

our setting it means that the only cases which lead to non-injective uniformly

quasiregular mappings are those in which the translation group T is isomorphic to

either Z '~ or Z '~- 1. A discussion o f the corresponding automorphic mappings can be

found in [MS], where such mappings are called respectively n- and ( n - 1)-periodic.

Proposition 3.1. Let F be a discrete group o f isometries o f Rn and let T, the

subgroup o f all translations ofF, be isomorphic to Z k. l f k E {1, ..., n - 2}, then there is no quasiregular mapping h : R n ~ ' ~ automorphic (in our sense) with respect to F.

P r o o f . Suppose h : IR '~ ~ is such a mapping and define h : It~ '~/T ~ by

o 7r = h, where 7r : R ' ~ Rn/T is the usual projection. The result o f Martio [Mo]

says that h must be infinite-to-one when k E {1, ..., n - 2}. But this contradicts the

fact that h is automorphic, since the group o f rotations R is finite and therefore

h-l(y) must also be finite for every y E h(R'~). []

3.2 Definition of Latt~s-type mappings Now fix a discrete group F

o f isometries o f R'~ (with translation group T isomorphic to Z n or Z '~-1) and a

corresponding automorphic mapping h : I~ n ~ ~ . To a given similarity A(z) =

AUz, with A > 1 and U E O(n) an orthogonal matrix, we will associate a mapping

f that solves Schr6der 's equation

(3.1) f o b = ho A.

This is possible whenever A satisfies the condition A o F o A -1 C F. Note that it is

also this condition that allows projecting the similarity A to a mapping Ma o f the

110 ~ MAYER

orbifold W~/F. In some sense, it is equivalent to consider either f or the mapping

MA on the orbifold.

Proposition- Defini t ion 3.2. Let A(x) = AUx, with A > 1 and U 6 0 (n) ,

be a similarity o f R n which satisfies the condition A o F o A -1 C F. Then there is a

uniformly quasiregular mapping f o f ~ --~ which satisfies Schrrder ' s equation (3.1).

We call a mapping f obtained in this manner a mapping o f Lattds type.

ProoL It is clear that there is a mapping f defined on h(R n) by (3.1), and it is

easy to check that f 6 UQR(h(E~)) . I fh(R n) =~-~, then we are done. This is the

case when h is n-periodic. The remaining case is when the translation group T is

isomorphic to Z n- 1. Then we can compactify the cylinder Z = lI~"/T by two points

-oo, co as in [MS]. If we choose coordinates so that the x,~-axis is orthogonal to

the translations of T, then the prime ends -oo, c~ correspond to x,~ = -oo and

x , = oo. It follows from [MS, w that there are points al, a2 6 ~ (which may

coincide) such that h(l~ n) =~'~\{al, a2} and the limits

lim h(x) = al and lim h(x) = a2

exist. From this we see that f extends to a continuous mapping of ~'~, and it

follows from Theorem 4.3 of [MRV] that this extension is uniformly quasiregular. []

3.3 Examples of Latt~s-type mappings We now discuss the possible

examples of Latt~s-type mappings.

The case k = n In this case the groups are crystallographic groups, the orbifolds

IR'~/F compact and the Julia set o f the corresponding Latt~s-type mappings is the

whole space ~~. Examples of this kind can be found in [My], where we have

given higher-dimensional analogues of Latt~s' chaotic rational function. The

automorphic mapping we used to obtain these examples is a quasimeromorphic

analogue of the Weierstrass P-function (see [MS]).

The case k = n - 1 Examples of Latt~s-type mappings in this situation are

the natural counterparts of power mappings from [My]. In these examples, the

two ends -oo, oo of the cylinder Z = R '~/T correspond to the completely invariant

fixed points of the mappings and form the exceptional set of these mappings.

There is one remaining case, namely, i f r has a rotation group R which identifies

the two prime ends. This corresponds to mappings having only one completely

invariant fixed point and a one-point exceptional set. In the following we give such


examples. They are natural analogues of Tchebychev polynomials. For simplicity,

we describe them only in the three-dimensional case.

T h e o r e m 2. There exist uniformly quasiregular mappings f o f ~ a with Julia

set Jy = D x {0}, where D denotes the unit disc in ~2, and whose Fatou set is the

basin o f attraction o f the superattracting completely invariant fuced point p = oo.

Proof . We first construct a quasiregular mapping h : 11~ 3 ~ I~ 3 automorphic

with respect to the discrete group of isometries F generated by the translations

x ~ x + 2el and x ~ x + 2e2 and by the two rotations x ~ ( - x l , x 2 , - x 3 ) and

Let Z =]0, l[2xR and Z- = Z n {x3 < 0}. From the construction of Zorich's

mapping (see [Ri]) we have a quasiconformal mapping h from the cylinder Z onto

the upper half-space H = {xa > 0} such that h ( Z - ) = H fq ]~3. But the half-ball

l~n •3 is conforrnally equivalent to the quarter-space {xl > 0, x3 > 0}, and in [V/i]

there is an explicit 4-quasiconformal mapping from the quarter-space onto the

half-space H. The composition of all these mappings gives a quasiconformal map

h : Z - ~ H. Up to a normalization by a M6bius transformation which preserves

IHI, we may suppose that it maps the prime end -oo of Z - to c~ and also that

h(Z- n {x3 = 0}) = D • {0}.

Next, we extend h to a quasiregular mapping of I~ 3 onto itself using reflections

on the faces of Z - and on al~ in the range. This extension, which we denote

still h, maps half-cylinders alternately onto the upper and lower half-space. By

construction, h is automorphic with respect to F (see also [Dr], where a similar

map has been described).

We can now apply Proposition-Definition 3.2: for any similarity A(x) = A U x

with A > 1 and U E 0(3) satisfying A o F o A -1 C F there is f E UQR(R 3)

solving Schrrder's equation (3.1). In particular, every mapping A(x) = dx, with

d = 2, 3 .... , induces such a uniformly quasiregular mapping f . The Julia set of

such a mapping is J f = h({x3 = 0}) = D x {0}. In its complement, the iterates ]k

converge locally uniformly to the completely invariant fixed point oo. Moreover,

infinity is superattracting since f is not locally homeomorphic at this point. []

Repelling cycles have a central role in the results of this paper. So we close

this section with the following observation.

P ropos i t ion 3.3. Repelling cycles are dense in the Julia set o f a Latt~s-type

mapping.

112 ~ MAYER

Proof . Let h: I~'~---,~ -~ be automorphic with respect to F, A(x) = AUx with

A > 1 and U E O(n) such that AoF o A -1 C F, and let f E UQR(~ n) be the induced

Latt~s-type mapping, i.e., the solution of (3.1).

Choose an open cube Q cl~ n with h(Q) D J( f ) and consider any ball B =

B(x,r) C Q with h(x) ~ J( f) . Since A is expanding, there is k ~ 1~ with

diamAkB > 2 diamQ. We have h o Ak(x) ~ J( f ) . Therefore, there is 3' ~ 1' with

,y o Ak(x) E Q, and this shows that "r o Ak(B) 3 Q D B. Consequently, the MSbius

transformation 3' o A k has a repelling fixed point x0 E B. The projection of this

point h(xo) c h(B) is a repelling fixed point of fk. We have shown that any open

set intersecting J( f ) contains a point of a repelling cycle of f . []

3.4 Applications to folded quasiregular mappings The explicit con-

struction of the quasiregular Tchebychev polynomials shows that such a mapping

preserves II~ 2 x {0}, and evidently it also preserves the Julia set J( f ) = D • {0}.

This allows us to consider the restriction of f to one of these two-dimensional sets;

it turns out that these restrictions are not quasiregular. They are, in fact, folded

quasiregular mappings, a name introduced by Srebro and Yakubov for a class of

mappings that are defined like quasiregular mappings except that they may change

orientation in some parts of the domain and preserve orientation in other parts (see

[SY]).

From the dynamical behaviour of the quasiregular Tchebychev polynomials we

get a result concerning compactness properties of folded quasiregular mappings. In

fact, let ~o be the restriction of one of the mappings of Theorem 2 to J( f ) = D x {0).

Then ~o is a folded quasiregular mapping of the unit disc onto itself, and we have

uniform control on the dilatation of all the iterates ~o k. One might call such a

mapping uniformly folded quasiregular. Since we consider the restriction of the

original mapping f to its Julia set, we obtain

C o r o n a r y 3.4. There exists a uniformly folded quasiregular mapping ~o : If) ----}

D such that the family {~o k, k E N) is not normal in any open subset o f D.

4 C o n f o r m a l s tructures

Up to now we have considered R'~ equipped with the standard euclidean

structure. An important tool in connection with uniformly quasiregular mappings

(and also uniformly quasiconformal groups) is to introduce conformal structures

with respect to which the mappings can be made "holomorphic". Good references

on this topic are [IM, Tu].


A conformal structure is a bounded measurable mapping # from ~ into the

space S(n) ofn x n real positive-definite matrices of determinant 1. If we forget for

a moment the boundedness and determinant 1 condition, then such a mapping # can

be viewed as a usual Riemannian metric with measurable coefficients. Since the

notion of conformality does not change if we rescale the metric, we can normalize

#. A canonical way to do this is to require that det #(x) = 1 for a.e. x E ~-~.

The boundedness condition is added in order to retain only the structures that are

adapted to quasiregular mappings, and means that there is a constant K _> 1 such

that

1 2 t X ~-'~. ~ [X[ ( /~(x) X < K[X[ 2 for all X E R n and a.e. x E

A conformal structure should be viewed as a measurable field of ellipsoids of

bounded eccentricity.

The usual pull-back of a Riemannian metric can be adapted to pull back a

conformal structure # by a quasiregular mapping f by defining

f*~(x) = f '(x)[# o f(x)] := tf'(x) #(f(x)) f'(x) for a.e. x E ~-~ det f'(x)2/n

Note that this pull-back behaves like the usual one under composition:

( f l o f 2 ) * , = f~(f;Iz).

A conformal structure is calledflat if there is a quasiregular mapping h with

h*# -- Id. Following Martin [Mn], we call a quasiregular mapping #-rational if

this mapping preserves the given conformal structure #, i.e.,

(4.1) #(x) = f*#(x) for a.e. x e ~ .

If f is/~-rational, then it is automatically uniformly quasiregular and # is said to

be f-invariant. In connection with these notions there are several important facts:

(i) In the planar case with # =Id the standard structure, (4.1) is precisely the

Cauchy-Riemann equation.

(ii) A mapping of the Sobolev class WI,'*(R n) is uniformly quasiregular if and

only if it is #-rational for some conformal structure #. In particular, for every

uniformly quasiregular mapping there is an invariant conformal structure.

(This result is due to Hinkkanen [Hi] when n = 2 and to Iwaniec-Martin

[IM] in all dimensions. See also [Tu] for the corresponding result concerning

quasiconformal groups.)

114 ~ MAYER

(iii) The measurable Riemann mapping theorem asserts that in two dimensions

every given conformal structure # can be pulled back by a quasiconformal

mapping ~ to the standard structure, i.e., ~*# =Id. In other words, every con-

formal structure o f ~ 2 is fiat. Together with (ii) we see that planar uniformly

quasiregular mappings are nothing other than quasiconformal conjugates of

holomorphic functions.

5 G e o m e t r y o f the c a n o n i c a l i n v a r i a n t c o n f o r m a l s t r u c t u r e

Let f be a Latt~s-type mapping and F c Isom(l~'~), h : / l ~ " ~ ~ \ E y the

corresponding group, quasiregular mapping respectively such that f o h = h o A for

some similarity A. It is easy to check that the (strong form of) automorphy of h

allows defining an f-invariant conformal struc~re # by taking the h-push-forward

of the standard structure:

(5.1) h'(x)[# o h(x)] = Id for a.e. x E II~ n.

We refer to this tt as the canonical conformal structure of f . In [My] we proved

that this canonical structure is essentially the only f-invariant conformal structure

in the case of the chaotic mappings. Here we analyse the geometry of such a

structure.

Proposition 5.1. Let # be the canonical conformal structure o f the Latt~s-

type mapping f and let g be any p-rational mapping. Then there is a similarity B

which projects on ~'~ /F so that g o h = h o B in R n.

In particular, any #-rational mapping is of Latt6s type. This is also in contrast

to the two-dimensional case. Rational functions "ofLatt6s type" can be perturbed

in the space of rational functions to mappings that are far from being Latt6s. For

example, this occurs in the quadratic family Pc(z) = z 2 + c where Po and P-2

are critically finite with parabolic orbifold. Another example is Lyubich's family

fa(z ) = 1 + A/z 2. In that case, the value A = - 2 corresponds to a Latt6s mapping.

The reason for this rigidity phenomenon is LiouviUe's Theorem. It would be

interesting to know whether or not Latt6s-type mappings are isolated in the whole

space of uniformly quasiregular mappings.

Proof . Consider the arcwise connected domain f~ = ~ \ h ( B h ) . Take x0 E R '~

with bo = g o h(xo) Ef t and choose H a branch of h -1 defined in a neighborhood

of b0. We can now consider the mapping ~o(x) = H o g o h(x) , which is well defined

in a neighborhood of x0. Since # is g-invariant and the h-pull-back of # is the


standard structure Id, Liouville 's Theorem implies that ~ is the restriction of a

M6bius transformation, still denoted ~ E M6b (~-n). I f we extend H along any arc

in f~, we get an extension of the map H o 9 o h. But this extension does always

coincide with the MSbius transformation qo. So we can write go = H o g o h in the

domain ft' = (9 o h) - l ( f l ) . Since ~-7 =~,~, it follows that g o h = h o ~ in I~ ~ . Now it

is clear that ~ fixes c~ and also that ~o projects on ll~n/F. The similarity we looked

for i s B = ~o. []

6 Characterization of Latt~s-type mappings

Here we prove the Main Theorem. The difficult part o f the proof is to establish

the sufficiency of the flatness condition concerning the f-invariant structure/z. In

preparing for this, we consider the case in which/~ is flat at a repelling fixed point

of f itself. So, in the following we suppose that the degree dOf > 2 and that #

is an f-invariant conformal structure fiat at a repelling fixed point o f f , which we

suppose to be the origin 0.

Let us first state a precise version of the fact that the flatness condition on #

yields flatness of the mapping f at the repelling fixed point.

L e m m a 6.1. There exists a quasiregular mapping h defined on I~ '~ with image

~ '~ \E S, f ix ing the origin, h(O) = O, which is homeomorphic in a neighborhood o f

0 and such that

(6.1) h'(x)[# o h(x)] = Id f o r a.e. x E R ~

and

(6.2) h o A(x) = f o h(x) fo r every x E ~,'~

where A(x) = AUx with A > I and U E O(n) an orthogonal matrix.

P roo f , By hypothesis, there is a quasiconformal mapping h defined in a

neighborhood W of 0 satisfying (6.1) for a.e. x E W. Consider the mapping

A(x) = h -1 o f o h(x) which is well-defined near 0. Since (6.1) is true near 0 and

since # is f-invariant, it follows that A is conformal and therefore, by Liouville 's

Theorem, is the restriction of a MSbius transformation. Therefore, up to normal-

ization by an auxiliary M6bius transformation, we may assume that A(x) = AUx

with U E O(n) and A > 1, since 0 is a repelling fixed point.

It remains to extend h. This can be done in the usual way using the dynamics

o f the mappings. I f x E R n choose k ~ 14 with A - k ( x ) E W . Then define

h(x) = f f o h o A -k (x ) .

116 ~ MAYER

Since (6.2) holds in A - i (W), this definition is independent o f the choice o f k. The

way h is defined and the fact that (6.1) and (6.2) are true near 0 implies that these

relations are true globally. In fact, choose for a given x ER '~ an integer k E N with

A-(k-1)(x) E W. Then we have

h o A(x) = f k o h o A - k ( A ( x ) ) = f o (fk-1 o h o A-(k- i ) ) (x) = f o h(x).

On the other hand, the f-invariance o f # implies

h'(z)[~ o h(z)] = (h o A - k ) ' ( x ) [(f~)'(h o A - k ( x ) ) [ # ( f k ( h o a-k(x)))]]

= (A-k ) ' ( x ) [h ' (A-k(x))[#(h(A-k(x)) )]]

= (A-k)'(x)[Id] = Id a.e. x E ~n.

Quasiregularity o f h is a consequence of the fact that the f k are uniformly quasireg-

ular. Finally, the range of this mapping is

h(Rn) = g f k ( h ( W ) ) = ~'~ \ E: . [] k>O

L e m m a 6.2. The mapping h: R n ~ n \ E $ obtained in Lemma 6.1 is infinite-

to-one. More precisely, f o r every y E ~ \ E I , card(h- i (y)) = c~.

P roo f . Let y E ~'~\(fk(Byk) U El ) . I f the degree o f f is d > 2, then there are

precisely d k distinct points ai with f k (a i ) = y. Let U = h(W) , where W is as in

the proof o f Lemma 6.1; then U c f ( U ) and we have an inverse h - i : U ---, W.

Since f ( U ) c f l+i(U), there is an index l with {ai, ...,aa~) C f ( U ) . In other

words, there are distinct points hi, ..., bd~ e U with f l + k ( b i ) = y. Consider now

x i = A l+k o h - l ( b i ) , i = 1, ..., d k. These are d ~ distinct h-preimages o fy .

Consequently, every point which is not in the set

h(Bh)UEf = g fk(Bf~ UEf), k>__i

a set of measure zero, has infinitely many preimages. Since h-i(y) is always a

discrete set it must be an infinite set for every y ~ ~ \ E f . []

The key of the proof of the Main Theorem is the following result.

Proposition 6.3. For any xi, x2 E IR '~ with h(xl) = h(x2) there is an isometry

3' E Isom(R '~) satisfying 7(xi) = x2 and h o "r = h.

P roo f . We first consider the situation x i , x2 E f~ = ~'~ \ h - l ( h ( B h ) ) . I f y =

h(xi) = h(x2) then we can choose H, a branch of h - i , defined in a neighborhood


of y such that H ( y ) = z2. Consider now the mapping H o h as a mapping defined

near Xl. It is conformal since h satisfies (6.1). By Liouville's Theorem, it is the

restriction of a MSbius transformation 7.

We now prove the ,y-invariance of h. The set f~ is a domain and hence arcwise

connected. Let zl E f~' = ,,/-1 (f~), z2 = 7(zl) and let a be an arc in f~ with endpoints

z2 and z2. Since h o c avoids h(Bh), the branch H can be extended to a neighborhood

of this arc. Using analytic continuation we conclude that 7Iv = H o hlu , where U

is a neighborhood of the arc 7 -1 o a. In particular, h(zx) = h(z2). Thus we have

proved that h o 7 = h on f~'. By continuity this is true in ~~.

Finally, we verify that 7 is an isometry. By Lemma 6.2, the set Ou = h - l ( y )

contains an infinite sequence {zk} of distinct points. We have seen that {7(zk)} c

Or. Since 0 u is discrete, these sequences both converge to infinity. Consequently

7 fixes infinity. I f 7 were not an isometry, then either 7 or ,y-1 would have an

attracting fixed point p in R '~. But this is impossible because it means that p is an

accumulation point o f the discrete set O u.

For the general case, x l , x 2 E ]~n, it suffices to approach xi by points xi,k E f~,

i = 1, 2. []

P r o o f o f the M a i n T h e o r e m . Consider first the case where f is o f Latt~s

t y p e , / T h e n the canonical conformal structure # defined in (5.1) is f-invariant.

Furthermore, # is flat in the complement o f h(Bh), which contains a point o f a

repelling cycle o f f (Proposition 3.3).

Let us prove the converse. Assume that f has an f-invariant conformal structure

# at a repelling fixed point o f an iterate fk. We first show that g = f k is of Latt~s

type. By Lemma 6.1, there is h : / ~ ' ~ ~~\Eg quasiregular satisfying (6.1) and

(6.2). Consider

F = {7 E Isom(l~'~); h o "7 = h).

By Proposition 6.3, F is transitive on every fiber O v = h - l ( y ) . Therefore, h is

automorphic with respect to F, and F is a discrete group since such a fiber Ov is

discrete. We have shown that g is o f Latt~s type.

Finally, observe that p is the canonical conformal structure o f g and that f is

#-rational. So Proposition 5.1 applies and shows that f is also o f Latt~s type. []

7 Regular i ty o f the conformal structure and its flatness

Here we seek a condition that implies the flatness o f an invariant conformal

structure. In the context o f quasiconformal groups, P. Tukia [Tu] showed the

following:

118 ~ MAYER

Let G be a uniformly quasiconformal group and # a G-invariant conformal

structure, l f # is approximately continuous at a radial limit point o f G, then G is

quasiconformally conjugate to a Mrbius group.

In fact, Tukia showed that in this context the conformal structure # is flat. We

use a local version of his method, where repelling fixed points play the role of

radial limit points to get the following non-injective version of his result. As a

Corollary we get Theorem 1.

T h e o r e m 3. Let G be a uniformly quasiregular semigroup o f ~ '~, n >_ 3, and

# a G-invariant conformal structure. Suppose that # is approximately continuous

at a repelling fixed point o f some non-injective element f E G. Then there is a

semigroup o f similarities { Ag; 9 E G} and a quasiregular mapping h defined on ~'~

such that g o h = h o Ag for every element 9 E G.

Proof . Suppose that 0 is a repelling fixed point of f with # approximately

continuous at 0. Let U be a neighborhood of 0 with U C f (U) and such that

the restriction of f to this neighborhood is injective. We suppose that B '~ C U.

Choose now sk > 0 maximal so that fk (sk ~'~) c ~'~. Then the quasiregular Montel

Theorem [Ri] implies that a subsequence of hk = fk o sk, which we consider as

mappings defined on 1~ '~, converges to a mapping h : B'~ ~ W = h(B '~) c ~'~.

By the choice of the sk and using usual distortion arguments, one shows that this

mapping cannot be constant. It is in fact a quasiconformal mapping; using the

hypothesis "# approximately continuous at 0", one shows as in [Tu, p. 339] that

h'(x)[# o h(x)] = #(0) for a.e. x E ~n.

It suffices now to compose h with an affme mapping in order to replace #(0) by

Id. This shows that # is flat at the origin. Our Main Theorem now says that f is of

Latt~s type. This, together with Proposition 5.1, completes the proof. []

[CH]

[Dr]

[Ge]

[Hi]

[HM]

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E T. Church and E. Hemmingsen, Light open maps on n-manifolds, Duke Math. J. 27 (1960), 527-536.

D. Drasin, On a method of Holopainen and Rickman, Israel J. Math. 101 (1997), 73-84.

F. W. Gehring, Rings andquasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393.

A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Aead. Sci. Fenn. Ser. A I Math. 21 (1996), 205-222.

A. Hinkkanen and G. Martin, Attractors in quasiregular semigroups, preprint.


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Volker Mayer U.R.A. 751

UFR DE MATHI~MATIQUES PURLS ET APPLIQU~'ES UNIVERSITI~ DE LILLE I

59655 VILLENEUVE D'ASCQ, CEDEX, FRANCE email: [email protected]

(Received July 10, 1997 and in revised form March 16, 1998)

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