「On the Iteration of Analytic Functions」(木村俊房先生の仕事から)

The paper examines some properties of the dynamics of entire functions which extend to general meromorphic functions and also some properties which do not. For a transcen- dental meromorphic function f(z) whose Fatou set F(f) has a component of connectivity at least three, it is shown that singleton components are dense in the Julia set J(f). Some problems remain open if all components are simply or doubly connected. Let I(f) denote the set of points whose forward orbits tend to ∞ but never land at ∞. For a transcendental meromorphic function f(z) we have J(f) = ∂I(f), I(f) ∩ J(f) 6 ∅. However in contrast to the entire case, the components of I(f) need not be unbounded, even if f(z) has only one pole. If f(z) has finitely many poles then, as in the entire case, F(f) has at most one completely invariant component.

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Dynamics of transcendental meromorphic functions

We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an unbounded component if and only if the function has no Baker wandering domains. We also give estimates of the Hausdorff dimension and the upper box dimension of the Julia set of a meromorphic function with a logarithmic singularity over infinity. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman-Valiron theory. The method is also applied to complex differential equations.

Dynamics of meromorphic functions with direct or logarithmic singularities

We show that for any transcendental meromorphic function f there is a point z in the Julia set of f such that the iterates f n (z) escape, that is, tend to ∞ , arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which f n (z) tends to ∞ at a bounded rate, and establish the connections between these sets and the Julia set of f . To do this, we show that the iterates of f satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a quasi-nested wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.

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Slow escaping points of meromorphic functions

On multiply-connected Fatou components in iteration of meromorphic functions ✩

An unbounded Fatou component U of a transcendental entire function is simply- connected. The paper studies the boundary behaviour of the Riemann map Ψ of the disc D to U , in particular the set Θ of ∂D where the radial limit of Ψ is ∞ . If U is not a Baker domain and ∞ is accessible in U , then Θ is dense in ∂D.I fU is a Baker domain in which f is not univalent, Θ contains a non-empty perfect subset of ∂D.E xamples show that Θ may be either countably infinite or residual in ∂D. The function f(z )= z + e −z leads to a component U with a particularly interesting prime end structure.

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Boundaries of unbounded fatou components of entire functions

The theory of Fatou and Julia is extended to include the dynamics of functions f which are meromorphic in \widehat{\mathbb{C}} outside a totally disconnected compact set E(f) at whose points the cluster set of f is \widehat{\mathbb{C}}. The Julia set is defined not only by the standard approach but is also characterized in terms of the set of points whose orbits approach a point of E(f). For the subclass where E(f) has a complement of class OAD and the inverse of f has a finite set of singular points it is shown that neither wandering components nor Baker domains occur in F(f)>. As an application, functions of a certain general class are shown to have a totally disconnected Julia set.

Dynamics of functions meromorphic outside a small set

For a transcendental entire function f we investigate the connectedness properties of the Julia set J(f) in the plane and in the Riemann sphere. We give examples where J(f) contains buried components, that is, components which do not meet the boundary of any component of the complement F(f) of J f). In connection with an old question of Fatou we show that if F f has a multiply-connected component, then J f has buried components which arc singletons. Such components are dense in J f).

Connectedness properties of julia sets of transcendental entire functions

In general the Julia set of an entire or meromorphic function is either connected or has an uncountable set of components. Conditions are found which ensure the second alternative. If a transcendental entire function has a completely invariant Fatou component then its Julia set is locally connected at no point in the plane. If the Julia set of a transcendental entire function is locally connected at some point, then the Julia set is connected. An entire function is constructed for which every periodic point is repelling, is a singleton component of the Julia set and lies on the boundary of no Fatou component.

Patricia Domínguez

Papers

Dynamics of transcendental meromorphic functions

Boundaries of unbounded fatou components of entire functions

Dynamics of functions meromorphic outside a small set

Connectedness properties of julia sets of transcendental entire functions

Some connectedness properties of julia sets