This paper is concerned with a generalisation of the classical theory of the dynamics associated to the iteration of a rational mapping of the Riemann sphere, to the more general setting of the dynamics associated to an arbitrary semigroup of rational mappings. We are partly motivated by results of Gehring and Martin which show that certain parameter spaces for KJeinian groups are essentially the stable basins of infinity for certain polynomial semigroups.Here we discuss the structure of the Fatou and Julia sets and their basic properties. We investigate to what extent Sullivan's 'no wandering domains' theorem remains valid. We obtain a complete generalisation of the classical results concerning classification of basins and their associated dynamics under an algebraic hypothesis analogous to the group-theoretical notion of 'virtually abelian'. We show that, in general, polynomial semigroups can have wandering domains. We put forward some conjectures regarding what we believe might be true. We also prove a theorem about the existence of filled in Julia sets for certain polynomial semigroups with specific applications to the theory of Kleinian groups in mind.
Let f and h be transcendental entire functions and let
g be a continuous and open map of the complex plane into itself with
g∘f=h∘g. We show that if f satisfies
a certain condition, which holds, in particular, if f has no wandering domains, then
g−1(J(h))=J(f). Here
J(·) denotes the Julia set of a function. We conclude that if f
has no wandering domains, then h has no wandering domains. Further, we show that
for given transcendental entire functions f and h, there are only
countably many entire functions g such that
g∘f=h∘g.
We investigate local dynamics of uniformly quasiregular mappings, give new examples and show in particular that there is no quasiconformal analogue of the Leau-Fatou linearization of parabolic dynamics.
An increasing homeomorphism / of the real line R onto itself is K-quasisymmetric if holds for all .v 6 R, t # 0. A K-quasisymmetric group is a group of K-quasisymmetric mappings under composition of functions.It is proved that if G is a K-quasisymmetric group, then there exists a quasisymmetric function / such that / " ' o G of is a group of linear functions.Pmc. London Math. Soc.(3). 51 (1985). 318 338.
UNIFORMLY QUASISYMMETRIC GROUPS3 1 9 / / G is not cyclic and contains a function without fixed points, then f is uniquely determined up to a linear transformation and q(f) ^ K. REMARK 1. In general, there are a variety of choices for /. More details will be provided in the course of the proof. REMARK 2. If fg are qs, then q{f~l) and q{fog) are bounded by numbers depending on q(f) and q(g) only [3]; a linear function is 1-qs. Hence, if/ is qs and if Gj is a group of linear functions, then / o G t o/" 1 is a uniformly quasisymmetric group. REMARK 3. It will turn out that if G is not cyclic and if all functions of G have a fixed point, then all of them fix the same point (and no other point). REMARK 4. If G is not cyclic and contains a function without fixed points, then the equality q(f) = K is possible. This happens, for example, if G = / o Gj o / " 1 , where f(x) = x for x ^ 0, f(x) = Kx for x ^ 0, K ^ 1, and G { is the group of all translations.In §2 we prove preliminary results (Lemmas 1-5), and in §3 we prove Theorem 1 for cyclic groups. In §4 we outline the proof in the general case, and in § § 5 and 6 we prove the lemmas required to complete the proof of Theorem 1. I wish to thank Professor W. K. Hayman for several helpful suggestions. I am also grateful to the referee for his comments, and in particular for showing how the proof of Theorem 1 for non-cyclic groups can be considerably shortened by making use of the compactness properties of quasisymmetric functions.
Preliminary resultsWe recall some distortion properties of qs functions. If L,-(x) = a,x + 6,, with a { > 0 and / = 1,2, if / is qs and g(x) = -/ ( -x), then [3]
An extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ → Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature (n = 4), then every quasiregular mapping f : M → M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π 1 -injective proper quasiregular mappings f : M → N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.
Abstract. Let f be a transcendental entire function of order less than 1/2. We introduce the method of "self-sustaining spread" to study the components of the set of normality of such a function. We give a new proof of the fact that any preperiodic or periodic component of the set of normality of f is bounded. We obtain the same conclusion for a wandering domain if the growth rate of f is never too small.
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