Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 31–39

BEHAVIOR OF QUASIREGULAR SEMIGROUPS NEAR ATTRACTING FIXED POINTS

Volker Mayer

Université de Lille I, U.R.A. 751, UFR de Mathématiques Pures et Appliquées F-59655 Villeneuve d’Ascq Cedex, France; mayer@gat.univ-lille1.fr

Abstract. We study the dynamical behavior of quasiconformal groups and of quasiregular semigroups near attracting fixed points. In the homeomorphic case we show that there is a 1parameter quasiconformal loxodromic group of \overline{R}^{n}, n ≥ 3, that cannot be quasiconformally linearized. This implies that Martin’s result, which says that the cyclic quasiconformal loxodromic groups are all quasiconformal conjugates of ⟨x ↦ 2x⟩, cannot be quantitative. In the non-injective situation we exhibit an example of a quasiregular semigroup whose elements have a common attracting fixed point and which cannot be quasiconformally linearized at this point. For the dynamics of uniformly quasiregular mappings at superattracting fixed points, we show that there are several distinct quasiconformal conjugacy classes by classifying the power mappings of [My2].

1. Introduction

A semigroup ℱ of uniformly quasiregular mappings is called a quasiregular semigroup. For invertible mappings, we consider quasiconformal groups, i.e. groups of uniformly quasiconformal mappings. In each case we consider semigroups whose elements are defined on \overline{R}^{n}, n ≥ 3, and have a common attracting fixed point. We are interested in the classification of the dynamics of ℱ at such a point by means of quasiconformal conjugacy.

The simplest quasiconformal groups whose elements have a common attracting fixed point are the cyclic loxodromic ones, that is G = ⟨g⟩ is loxodromic if there are distinct points p, q ∈ \overline{R}^{n} so that either g^{k} or g^{−k} converges uniformly on compact sets of \overline{R}^{n} \\{q\} to p as k → ∞.

Theorem (Martin). A cyclic quasiconformal group generated by a loxodromic element is quasiconformally conjugate to the Möbius group generated by x ↦ 2x.

In other words, up to quasiconformal conjugacy, all the cyclic quasiconformal groups are identical. For general uniformly quasiregular mappings, there is the following corresponding local result.

Theorem (Hinkkanen and Martin). Suppose that f is a uniformly quasiregular mapping of \overline{R}^{n} having an attracting fixed point p. Then there is a quasiconformal mapping h of \overline{R}^{n} with h(p) = 0 and such that h ∘ f ∘ h^{−1}(x) = \frac{1}{2}x in a neighborhood of the origin 0.

1991 Mathematics Subject Classification: Primary 30C65; Secondary 58F23.

Our first aim is to establish sharpness of these results. Their proofs are based on deep topological facts that do not allow one to get quantitative results. More precisely, it was not known if the conjugating mapping can be chosen to have a dilatation depending only on the dilatation of the semigroup and possibly the dimension. We prove that this is not true in general and that these results cannot be extended to the non-discrete case: there are examples of non-discrete uniformly quasiregular semigroups which are not locally quasiconformally conjugate at a repelling fixed point to the corresponding conformal model.

We also consider the case of superattracting fixed points. These are fixed points which are at the same time branch points. Note that in this case there is no conformal model since 1-quasiregular mappings have no branch points. In [My2] we gave examples of uniformly quasiregular mappings having superattracting fixed points. Here we classify them and, in particular, we show that it is possible to choose two of them having the same degree and which are not, locally at the superattracting fixed points, in the same quasiconformal conjugacy class.

2. Definition of quasiregular mappings and notations

Let D ⊂ R^{n} be a domain and f: D → R^{n} a mapping of Sobolev class W_{loc}^{1,n}(D). We consider only orientation preserving mappings, thus the Jacobian determinant J_{f}(x) ≥ 0 for a.e. x ∈ D. Such a mapping is said to be K -quasiregular, where 1 ≤ K < ∞, if

max_{|h|=1} |f′(x)h| ≤ K min_{|h|=1} |f′(x)h| for a.e. x ∈ D.

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 be continuous, open and discrete, and we shall always assume that this has been done. If D is a domain of the compactification \overline{R}^{n} (equipped with the spherical metric; thus \overline{R}^{n} is isometric via stereographic projection with the n-sphere S^{n}), then we use the chart at infinity x ↦ x/|x|^{2} to extend, in the obvious manner, the notion of quasiregularity to mappings f: D → \overline{R}^{n}. Such mappings are said to be quasimeromorphic. The branch set B_{f} is the set of points x ∈ D for which f is not locally homeomorphic at x. The notation i(x, f) stands for the local index of f at x, as defined in [R, p. 18].

A mapping f of a domain D into itself is called uniformly quasiregular if there is some 1 ≤ K < ∞ such that all the iterates f^{k} are K -quasiregular. We abbreviate this by f ∈ UQR(D). Quasiconformal groups and quasiregular semigroups are (semi-) groups whose elements are all K -quasiconformal, K -quasi-regular respectively, for some fixed K ≥ 1.

3. The quasiconformal group case

We consider here cyclic and 1-parameter quasiconformal loxodromic groups. Using an example of Tukia [T] we find:

Theorem 1. For any K > 1, there exists a 1-parameter quasiconformal loxodromic group that cannot be conjugate by a quasiconformal mapping to a Möbius group.

Before proving this, let us first deduce the announced distortion result in the discrete case. Given G a cyclic quasiconformal loxodromic group, we denote by K(G) the smallest possible dilatation K ≥ 1 for which there exists a K -quasi-conformal mapping h with h ∘ G ∘ h^{−1} a Möbius group. By Martin’s result, K(G) < ∞.

Corollary 3.1. For any K > 1, there exists a sequence of cyclic K -quasi-conformal loxodromic groups G_{k} with lim_{k→∞} K(G_{k}) = ∞.

Proof. Let K > 1 and G = {g_{t} : t ∈ R} be a 1-parameter K -quasiconformal loxodromic group which cannot be quasiconformally conjugate to a Möbius group. Such a group exists by Theorem 1. Denote by G_{k} the cyclic subgroup of G generated by g_{2^{−k}}. This is an increasing sequence of subgroups exhausting G. Suppose now that there is H > 1 and, for any k ≥ 1, a H -quasiconformal mapping h_{k}, normalized so that it fixes 0, 1, ∞, with h_{k} ∘ G_{k} ∘ h_{k}^{−1} ⊂Möb( \overline{R}^{n} ). The normalization implies that there is a uniformly convergent subsequence h_{k_{j}} with limit h a quasiconformal mapping. We have now a contradiction since this limit mapping conjugates G to a subgroup of Möb(\overline{R}^{n}): let g ∈ G and choose g_{t(k_{j})} ∈ G_{k_{j}} such that g_{t(k_{j})} converges uniformly to g. Then we see that

φ = h ∘ g ∘ h^{−1} = lim_{j→∞} h_{k_{j}} ∘ g_{t(k_{j})} ∘ h_{k_{j}}^{−1}

is a Möbius transformation.

We give now a version of Tukia’s group, the main ingredient for the proof of Theorem 1. Let f be a quasiconformal mapping of R^{2} such that f_{|R} is bihölder α with α ∈]\frac{1}{2}, 1[: there is c ≥ 1 with

(1)  \frac{1}{c}|x − y|^{α} ≤ |f(x) − f(y)| ≤ c|x − y|^{α}  for all x, y ∈ R.

So Γ = f(R) is a 1/α-dimensional fractal curve. In Tukia’s example it is the Von Koch Snowflake curve and, more generally, one can use other fractal bilipschitz homogeneous curves [My1]. The mapping f can be chosen to have a dilatation K tending to 1 when α → 1 [Mc]. The 1-parameter quasiconformal loxodromic group we look for is the conjugate of the group

H = {x = (u, v) ∈ R^{2} × R^{n−2} ↦ h_{t}(x) = (e^{t}u, e^{αt}v) : t ∈ R}

by the mapping F = f × Id: R^{2} × R^{n−2} → R^{2} × R^{n−2} :

(2)  G = F ∘ H ∘ F^{−1} = {g_{t} = F ∘ h_{t} ∘ F^{−1} : t ∈ R}.

Proof of Theorem 1. Let u ∈ Γ, v_{0} ∈ R^{n−2} and p_{0} = (u, v_{0}). Consider the segment σ = [(u, 0), p_{0}]. For p = (u, v) we have g_{t}(p) = (f(e^{t}f^{−1}(u)), e^{αt}v). From this and from the bihölder α inequality (1) follows that there exists t_{0} > 0 and C ≥ 1 such that

(3)  \frac{|t|^{α}}{C} ≤ |g_{t}(p) − p| ≤ C|t|^{α} for every p ∈ σ and 0 ≤ |t| ≤ t_{0}.

Suppose now that there is a quasiconformal mapping h conjugating G to a subgroup of Möb(\overline{R}^{n}). After normalization we have that h is a mapping of \overline{R}^{n} fixing 0 and that the conformal group Φ = h ∘ G ∘ h^{−1} has the form Φ = {φ(x) = e^{t}U_{t}x : t ∈ R} with U_{t} ∈ O(n). Evidently, the orthogonal matrices U_{t} depend continuously on t and U_{0} = Id. Since the conjugation induces an isomorphism of groups, there is r ≠ 0 so that h ∘ g_{t} = φ_{rt} ∘ h.

Take p ∈ σ, t ∈ [−t_{0}, 0] and let a ∈ R^{n} be any point with |a| = |g_{t}(p) − p|. Since h is quasiconformal, we have

|h(p + a) − h(p)| ≲ |h(g_{t}(p)) − h(p)| = |φ_{rt} ∘ h(p) − h(p)|,

where ≲ stands for “≤ up to a constant depending only on n and the dilatation constant of h”. If t_{0} is sufficiently small, this can be majorized by

|h(p + a) − h(p)| ≲ |e^{rt} − 1| |h(p)| ≲ |th(p)|.

On the other hand, |t| ≤ (C|a|)^{1/α} in view of inequality (3). Therefore

lim_{a→0} |h(p + a) − h(p)|/|a| = 0

for all p ∈ σ. In other words, h restricted to σ is constant which is impossible.

4. Attracting fixed points

For the notion of an attracting fixed point we use the definition of [HM]: a fixed point p of f is called attracting if there is a topological ball U with \overline{f(U)} ⊂ U and that f_{|U} is injective.

The proof of Theorem 1 is entirely local and thus these groups cannot be made conformal by a quasiconformal change of coordinates in any neighborhood of the fixed point 0. Therefore, if the restriction of a quasiregular semigroup to a neighborhood of an attracting fixed point (which in our cases will always be common to all the elements of the semigroup) behaves like the quasiconformal wild groups of Theorem 1 around 0, then this semigroup cannot be quasiconformally linearized at this point. It turns out that we can construct such a semigroup as follows.

Theorem 2. There exists a quasiregular semigroup ℱ generated by two mappings, with 0 a common attracting fixed point for all the mappings f ∈ ℱ and for which there is no quasiconformal mapping defined in a neighborhood W of 0 conjugating the restriction to W of the elements of ℱ to conformal mappings.

Note that the restriction of ℱ to a neighborhood of 0 is an abelian semigroup with two generators, so in some sense our example is as close as possible to the cyclic case.

We now explain how to get such a semigroup. Let f be a uniformly quasiregular mapping of \overline{R}^{n} with attracting fixed point p. The existence of nonhomeomorphic such mappings can be found in [IM]. Hinkkanen and Martin’s result asserts that there is h ∈ QC(\overline{R}^{n}) with

h ∘ f ∘ h^{−1}(x) = \frac{1}{2}x in some neighborhood W of 0.

We set f̃ = h∘f ∘h^{−1} . Let G be the quasiconformal group of (2) and g = g_{−2} ∈ G. This time we use Martin’s linearization result to get H ∈ QC( \overline{R}^{n} ) conjugating g:

H^{−1} ∘ g ∘ H(x) = \frac{1}{2}x for all x ∈ \overline{R}^{n}.

The first generator of our semigroup will be

f_{2} = (H ∘ h) ∘ f ∘ (H ∘ h)^{−1} = H ∘ f̃∘ H^{−1}.

The second generator will behave like the mapping g_{−3} of G near 0 and be identically f_{2} outside some ball. To obtain it we modify the construction of g_{−2} . For this we suppose that the neighborhood W satisfies H(W) ⊃ F (B(0, 1)) where F is the mapping of (2). Let r: R^{n} → [−3, −2] be smooth with r ≡ −2 in |x| > 1, r ≡ −3 in

h_{−2}(B(0, 1)) = {(e^{−2}u, e^{−2α}v) : x = (u, v) ∈ B(0, 1)}

and so that the restriction of r to any ray τ_{y} = {sy : s > 0}, y ∈ S^{n−1} , is increasing. The conjugate of the mapping x = (u, v) ↦ φ(x) = (e^{r(x)}u, e^{αr(x)}v) by the mapping F of (2), so the mapping g̃ = F ∘ φ ∘ F^{−1} , is uniformly quasiconformal and loxodromic. It also admits a quasiconformal conjugacy H̃, that we need to construct explicitly starting from the conjugacy H of g: since g(x) = g̃(x) for x ∈ Ω′ = F ({|x| > 1}) we can define H̃ = H in H^{−1}(Ω′). For x ∉ H^{−1}(Ω′) choose k ∈ N so that 2^{k}x ∈ H^{−1}(Ω′). Then define H̃(x) = g̃^{k} ∘ H ∘ (2^{k}x). Now it is clear that H̃ is a quasiconformal mapping with

H̃^{−1} ∘ g̃ ∘H̃(x) = \frac{1}{2}x for all x ∈ R^{n}.

The mapping we are looking for is

f_{3} = (H̃ ∘ h) ∘ f ∘ (H̃ ∘ h)^{−1} =H̃ ∘ f̃∘H̃^{−1}.

Consider now the semigroup ℱ = ⟨f_{2}, f_{3}⟩ generated by these mappings. By construction, f_{j} ≡ g_{−j} in Ω = F ∘ h_{−2}(B(0, 1)), j = 2, 3, where the g_{−j} are the elements of the group G. This gives the following elementary properties of ℱ:

Lemma 4.1. The semigroup ℱ is a quasiregular semigroup and its restriction to Ω is an abelian semigroup generated by g_{−2} and g_{−3} .

Proof. We show that any mapping f ∈ ℱ is K^{3} -quasiregular, where K is supposed to be the dilatation constant common to f_{2} , f_{3} and all its iterates. Let x ∈ \overline{R}^{n}, y = f(x) and consider the following points which correspond to a (not necessarily unique) decomposition of f,

(4)  x_{0} = x, x_{1}, …, x_{k} = y  where x_{l} = f_{2}(x_{l−1}) or x_{l} = f_{3}(x_{l−1}).

Then we have two cases: either all the x_{l} ∈ Ω′, l = 0, …, k, and in that case we have a dilatation K of f at x due to the fact that H ≡ H̃ in H^{−1}(Ω′). In the other case there is a first point x_{j} outside Ω′. If j = k, then the dilatation of f at x is still K since we can choose in (4) always the same mapping f_{2} or f_{3} for every l = 0, …, k − 1. If j < k − 1, then we proceed as follows. The condition H(W) ⊃ F (B(0, 1)) implies f_{2} = g_{−2} and f_{3} = g̃ in ^{c}Ω′. Since g(^{c}Ω′) = g̃(^{c}Ω′) = Ω, necessarily x_{l} ∈ Ω for all l > j. In Ω the mappings coincide with the mappings g_{−2}, g_{−3} of the (uniformly) quasiconformal group G. Therefore we have in this last case a dilatation K for the composition of the mappings that map x_{0} to x_{j}, a factor K when x_{j} is mapped to x_{j+1} and one more dilatation factor K for the remaining step when x_{j+1} is mapped to x_{k}.

Proof of Theorem 2. We prove by contradiction that the semigroup ℱ = ⟨f_{2}, f_{3}⟩ described above cannot be quasiconformally linearized at 0. Suppose that there is h a quasiconformal mapping defined on a neighborhood W of 0 such that h ∘ f_{j}_{|W} ∘ h^{−1} is the restriction of a Möbius transformation φ_{j}, j = 2, 3. Choose ϱ ∈]0, e^{−2}[ so that U = F (B(0, ϱ)) ⊂ W. Then f_{j}_{|U} = g_{−j}_{|U} for j = 2, 3 and

\overline{g_{t}(U)} ⊂ U for all t < 0. Since the f_{j} are commuting in U, φ_{2} and φ_{3} must have the same fixed points. The group Φ = \overline{⟨φ_{2}, φ_{3}⟩} is a 1-parameter loxodromic group since it cannot be discrete. It follows that

h ∘ {g_{t}_{|U} : t < 0} ∘ h^{−1} ⊂ Φ_{|h(U)}

which is impossible in view of the proof of Theorem 1.

5. Superattracting fixed points

A fixed point of a uniformly quasiregular mapping f which is also a branch point is called a superattracting fixed point. Recall that f is attracting at such a point, an immediate consequence of the local behavior of quasiregular mappings; see [R], [My2].

5.1. Counterparts of power mappings. In [My2] we gave examples of uniformly quasiregular mappings of \overline{R}^{3} having superattracting fixed points.

These mappings are the natural counterparts of the power mappings and they are obtained in a similar manner: Let h: R^{3} → R^{3} \ {0} be Zorich’s mapping (see [R] for a construction). This is the three-dimensional counterpart of the exponential mapping and is characterized by the fact that it is automorphic with respect to the group of isometries

Γ = ⟨(z, t) ↦ (−z, t) : (z, t) ↦ (z + 2, t) : (z, t) ↦ (z + 2i, t)⟩.

The analogue of the power mappings are the solutions f_{A} of Schröder’s equation

(5)  f_{A} ∘ h = h ∘ A

where A(z, t) = (λz, |λ|t) with λ ∈ C, |λ| > 1 and where we identified x ∈ R^{3} with (z, t) ∈ C × R. This conformal mapping A has to satisfy A ∘ Γ ∘ A^{−1} ⊂ Γ in order to get a well defined solution f_{A} of Schröder’s equation. The superattracting fixed points of these mappings are 0 and ∞.

Note that to each such mapping f_{A} there corresponds a Lattès rational mapping φ_{A} which is again a solution of a Schröder equation:

φ_{A} ∘ ℘(z) = ℘(λz), z ∈ C,

where ℘ is the Weierstrass P -function. Moreover, after a quasiconformal change of coordinates, we may suppose that the restriction of f_{A} to the two-sphere S^{2} is precisely φ_{A} [My2].

5.2. Classification. As in the attracting case we are interested in the quasiconformal classification of the dynamics of uniformly quasiregular mappings near superattracting fixed points. In contrast to the attracting case there is no conformal model, since Liouville’s theorem asserts that in the conformal case there can be no branching. We classify the above power mappings and deduce in particular that there cannot be a unique model.

Since we are dealing with quasiconformal conjugacy we have to respect a topological invariant, namely the local index. Two mappings f_{1} , f_{2} can only be locally conjugate near their superattracting fixed points p_{1} , p_{2} respectively, if they have the same local index there: i(p_{1}, f_{1}) = i(p_{2}, f_{2}). Hence conjugate power mappings must have the same degree given by |λ|^{2} .

Theorem 3. If f_{A} and f_{B} are two solutions of Schröder’s equation (5) having the same degree, then there is a local quasiconformal conjugacy between these mappings near their superattracting fixed point 0 if and only if A = B.

This result shows in particular that there are mappings f_{A} and f_{B} having the same degree but for which there is no local quasiconformal conjugacy relating them near their superattracting fixed point 0.

It suffices to choose A(z, t) = (dz, dt) and B(z, t) = (idz, dt), d ∈ {2, 3, …}. Since there is no conformal model and motivated by the particular construc-

tion of the above power mappings—which gives in dimension n = 2 precisely the mappings z ↦ z^{d} —we are led to think that these power mappings may in fact be the standard models for the local behavior of uniformly quasiregular mappings at their superattracting fixed points. We do not know any examples of a different kind.

Proof. When we express Zorich’s mapping by h(z, t) = (r, ξ), where (r, ξ) ∈ R^{+} × S^{2} are spherical coordinates, then r = e^{t} and ξ = ξ(z) is independent of t. Due to this special form we see that f_{A}(r, ξ) = (r^{|λ|}, g_{A}(ξ)), with g_{A} ∈ UQR(S^{2}) and, moreover, up to a quasiconformal change of coordinates, g_{A} is the Lattès mapping φ_{A} corresponding to f_{A}.

Suppose now that there is a quasiconformal mapping Φ defined near 0 with f_{A} ∘ Φ = Φ ∘ f_{B} still near the origin. We show that this implies that φ_{A} and φ_{B} are equivalent, which means that there are θ_{1} , θ_{2} homeomorphisms that are isotopic rel P_{φ_{B}}, the post critical set of φ_{B}, with θ_{1} ∘ φ_{B} = φ_{A} ∘ θ_{2} (cf. [DH]).

First we show that

(6)  |Φ(r, ξ)| = s_{r},  with s_{r} independent of ξ ∈ S^{2}.

If this is not true, then there is r > 0, ξ_{1}, ξ_{2} ∈ S^{2} and s_{1} < s_{2} with |Φ(r, ξ_{1})| = s_{1} < s_{2} = |Φ(r, ξ_{2})|. Then it follows that

f_{A}^{k}(Φ(r, ξ_{i})) = Φ(f_{B}^{k}(r, ξ_{i})) = Φ(r^{|λ|^{k}}, g_{B}^{k}(ξ_{i}))

and, when we set (s_{i}, η_{i}) = Φ(r, ξ_{i}),

f_{A}^{k}(Φ(r, ξ_{i})) = f_{A}^{k}(s_{i}, η_{i}) = (s_{i}^{|λ|^{k}}, g_{A}^{k}(η_{i})).

Therefore, for every k ∈ N there is a_{i,k} = g_{B}^{k}(ξ_{i}) for which

\frac{|Φ(r^{|λ|^{k}}, a_{1,k})|}{|Φ(r^{|λ|^{k}}, a_{2,k})|} = (\frac{s_{1}}{s_{2}} )^{|λ|^{k}} ⟶ 0 (k → ∞).

But this conflicts with our assumption of the quasiconformality of Φ.

Equation (6) means that Φ(rS^{2}) = s_{r}S^{2} . Set Φ_{r}(ξ) = Φ(r, ξ). This is a quasiconformal mapping of S^{2} with

g_{A} ∘ Φ_{r} = Φ_{r^{|λ|}} ∘ g_{B}.

It is clear that Φ_{r}(C_{g_{B}}) = C_{g_{A}} and the post critical set is also fixed by these mappings: Φ_{r}(P_{g_{B}}) = P_{g_{A}}. This implies that g_{A} and g_{B} are equivalent and therefore the same is true for the associated Lattès mappings φ_{A}, φ_{B}. The assertion A = B follows.

Behavior of quasiregular semigroups near attracting fixed points 39

References

[DH] Douady, A., and J.H. Hubbard: A proof of Thurston’s topological characterization of rational functions. - Acta Math. 171, 1993, 263–297.

[HM] Hinkkanen, A., and G. Martin: Attractors in quasiregular semigroups. - Proceedings of the XVI Nevanlinna colloquium, edited by I. Laine and O. Martio, de Gruyter, Berlin–New York, 1996, 135–141.

[IM] Iwaniec, T., and G. Martin: Quasiregular semigroups. - Ann. Acad. Sci. Fenn. Math. 21, 1996, 241–254.

[R] Rickman, S.: Quasiregular Mappings. - Springer-Verlag, 1993.

[Mc] McKemie, M.J.: Quasiconformal groups with small dilatation. - Ann. Acad. Sci. Fenn. Math 12, 1987, 95–118.

[My1] Mayer, V.: Trajectoires de groupes à 1-paramètre de quasi-isométries. - Rev. Mat. Iberoamericana 11, 1995, 143–164.

[My2] Mayer, V.: Uniformly quasiregular mappings of Lattès’ type. - Conform. Geom. Dyn. 1, 1997, 104–111.

[T] Tukia, P.: A quasiconformal group not isomorphic to a Möbius group. - Ann. Acad. Sci. Fenn. Math. 6, 1981, 149–160.

Received 15 January 1998