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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
QUASIREGULAR ANALOGUES 109
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
QUASIREGULAR ANALOGUES 111
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].
QUASIREGULAR ANALOGUES 113
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
QUASIREGULAR ANALOGUES 115
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
QUASIREGULAR ANALOGUES 117
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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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)
