Multiply Connected Wandering Domains Research Articles

Multiply connected wandering domains of meromorphic functions: the pursuit of uniform internal dynamics

AbstractRecently, Benini et al showed that, in simply connected wandering domains of entire functions, all pairs of orbits behave in the same way relative to the hyperbolic metric, thus giving us our first insight into the general internal dynamics of such domains. The author proved in a recent paper [G. R. Ferreira. Multiply connected wandering domains of meromorphic functions: internal dynamics andconnectivity. J. Lond. Math. Soc. (2) 106 (2022), 1897–1919] that the same is not true for multiply connected wandering domains, a natural question is how inhomogeneous multiply connected wandering domains can be. We give an answer to this question, in that we show that uniform dynamics inside an open subset of the domain generalizes to the whole wandering domain. As an application of this result, we construct the first example of a meromorphic function with a semi-contracting infinitely connected wandering domain.

Multiply connected wandering domains of meromorphic functions: Internal dynamics and connectivity

We discuss how the nine-way classification scheme devised by Benini et al. for the dynamics of simply connected wandering domains of entire functions, based on the long-term behaviour of the hyperbolic distance between iterates of pairs of points and also the distance between orbits and the domains' boundaries, carries over to the general case of multiply connected wandering domains of meromorphic functions. Most strikingly, we see that not all pairs of points in such a wandering domain behave in the same way relative to the hyperbolic distance, and that the connectivity of the wandering domain greatly influences its possible internal dynamics. After illustrating our results with the well-studied case of Baker wandering domains, we further illustrate the diversity of multiply connected wandering domains in general by constructing a meromorphic function with a wandering domain without eventual connectivity. Finally, we show that an analogue of the "convergence to the boundary" classification of Benini et al. does hold in general, and add new information about how this convergence takes place.

Classifying simply connected wandering domains

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

Eremenko points and the structure of the escaping set

Much recent work on the iterates of a transcendental entire function f f has been motivated by Eremenko’s conjecture that all the components of the escaping set I ( f ) I(f) are unbounded. We prove several general results about the topological structure of I ( f ) I(f) including the fact that if I ( f ) I(f) is disconnected, then it contains uncountably many pairwise disjoint unbounded continua, all of which are subsets of the fast escaping set. We give analogous results for the intersection of I ( f ) I(f) with the Julia set when multiply connected wandering domains are not present, and show that completely different results hold when such wandering domains are present. In proving these, we obtain the unexpected result that some types of multiply connected wandering domains have complementary components with no interior, indeed uncountably many.

Hyperbolic Metric and Multiply Connected Wandering Domains of Meromorphic Functions

AbstractIn this paper, in terms of the hyperbolic metric, we give a condition under which the image of a hyperbolic domain of an analytic function contains a round annulus centred at the origin. From this, we establish results on the multiply connected wandering domains of a meromorphic function that contain large round annuli centred at the origin. We thereby successfully extend the results of transcendental meromorphic functions with finitely many poles to those with infinitely many poles.

Permutable entire functions and multiply connected wandering domains

Let f and g be permutable transcendental entire functions. We use a recent analysis of the dynamical behavior in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of f and g are equal; in particular, we show that J(f)=J(g) provided that neither f nor g has a simply connected wandering domain in the fast escaping set.

ON THE UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS

AbstractWe investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.

On multiply connected wandering domains of meromorphic functions

AbstractWe describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou components of transcendental meromorphic functions. We also show that if f is meromorphic, U is a bounded component of F(f) and V is the component of F(f) such that f(U)⊂V, then f maps each component of ∂ U onto a component of the boundary of V in ℂ̂. We give examples which show that our results are sharp; for example, we prove that a multiply connected wandering domain can map to a simply connected wandering domain, and vice versa.

Weakly repelling fixed points and multiply-connected wandering domains of meromorphic functions

We consider the dynamics of a transcendental meromorphic function f(z) with only finitely many poles and prove that if f has only finitely many weakly repelling fixed points, then there is no multiply-connected wandering domain in its Fatou set.

ON WANDERING AND BAKER DOMAINS OF TRANSCENDENTAL ENTIRE FUNCTIONS

Let f denote a transcendental entire function, and I(f), I0(f), T(f) and A(f) be denoted as follows: [Formula: see text][Formula: see text] Let D denote a Fatou component of F(f). We have established the relationships between D and I(f), I0(f), T(f) or A(f), when D is a Baker domain or a multiply-connected wandering domain or a simply-connected infinitely wandering domain.

On the connectivity of Julia sets of transcendental entire functions

We have two main purposes in this paper. One is to give some sufficient conditions for the Julia set of a transcendental entire function to be connected or to be disconnected as a subset of the complex plane . The other is to investigate the boundary of an unbounded periodic Fatou component , which is known to be simply-connected. These are related as follows: let from a unit disk , then under some mild conditions we show that the set of all angles where is dense in is an attracting basin, a parabolic basin or a Siegel disk. If is not univalent, then or at least its closure contains a certain perfect set, which means the boundary has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set . If is a Baker domain on which is univalent, however, we shall show by giving an example that , which has a rather simple structure, and, moreover, in the Riemann sphere is connected if and only if has no multiply-connected wandering domains.

Weakly repelling fixpoints and the connectivity of wandering domains

It is proved that if a transcendental meromorphic functionffhas a multiply-connected wandering domain, thenffhas a fixpointz0z_0such that|f′(z0)|>1|f’(z_0)|>1orf′(z0)=1f’(z_0)=1. Entire functions with a multiply-connected wandering domain have infinitely many such fixpoints. These results are used to show that solutions of certain differential equations do not have wandering domains at all.

Some entire functions with multiply-connected wandering domains

AbstractA component U of the complement of the Julia set of an entire function ƒ is a wandering domain if the sets ƒn(U) are mutually disjoint, where n ∈ℕ and ƒn is the n-th iterate of ƒ. Examples are given of entire ƒ of order , which have multiply-connected wandering domains. An example is given where the connectivity is infinite.