Stable components in the parameter plane of transcendental functions of finite type

We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call {\em shell components}, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the {\em virtual center}, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of finite type.Comment: 41 pages, 13 figure

On the connectivity of the Julia sets of meromorphic functions

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton's method for entire maps are simply connected, which solves a well-known open question.Comment: 34 pages, 10 figure

A separation theorem for entire transcendental maps

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p-$periodic point or one parabolic immediate basin invariant under $f^p$. This result generalizes the Goldberg-Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel discs; that "hidden components" of a bounded Siegel disc have to be either wandering domains or preperiodic to the Siegel disc itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values

Singular values and bounded Siegel disks

Let f be an entire transcendental function of finite order and Δ be a forward invariant bounded Siegel disk for f with rotation number in Herman's class H. We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follo

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

Deformation of Entire Functions with Baker Domains

We consider entire transcendental functions f with an invariant (or periodic) Baker domain U. First, we classify these domains into three types (hyperbolic, simply parabolic and doubly parabolic) according to the surface they induce when we quotient by the dynamics. Second, we study the space of quasiconformal deformations of an entire map with such a Baker domain by studying its Teichmüller space. More precisely, we show that the dimension of this set is infinite if the Baker domain is hyperbolic or simply parabolic, and from this we deduce that the quasiconformal deformation space of f is infinite dimensional. Finally, we prove that the function f(z) = z + e −z, which possesses infinitely many invariant Baker domains, is rigid, i.e., any quasiconformal deformation of f is affinely conjugate to f.

Arnold Disks and the Moduli of Herman Rings of the Complex Standard Family

We consider the Arnold family of analytic diffeomorphisms of the circle x ↦ → x + t + a 2π sin(2πx) mod (1), where a, t ∈ [0, 1) and its complexification fλ,a(z) = λze a 1 2 (z − z), with λ = e2πita holomorphic self map of C ∗. The parameter space contains the well known Arnold tongues Tα for α ∈ [0, 1) being the rotation number. We are interested in the parameters that belong to the irrational tongues and in particular in those for which the map has a Herman ring. Our goal in this paper is twofold. First we are interested in studying how the modulus of this Herman ring varies in terms of the parameter a, when a tends to 0 along the curve Tα. We survey the different results that describe this variation including the complexification of part of the Arnold tongues (called Arnold disks) which leads to the best estimate. To work with this complex parameter values we use the concept of the twist coordinate, a measure of how far from symmetric the Herman rings are. Our second goal is to investigate the slice of parameter space that contains all maps in the family with twist coordinate equal to one half, proving for example that this is a plane in C2. We show a computer picture of this slice of parameter space and we also present some numerical algorithms that allow us to compute new drawings of non–symmetric Herman rings of various moduli