arXiv subjects

Gwyneth M. Stallard

Publications and source records attributed to Gwyneth M. Stallard.

17 recordsLinked to original sources

arXiv2024

Shrinking targets and recurrent behaviour for forward compositions of inner functions

We prove sharp results about recurrent behaviour of orbits of forward compositions of inner functions, inspired by fundamental results about iterates of inner functions, and give examples to illustrate behaviours that cannot occur in the simpler case of iteration. A result of Fernández, Melián and Pestana gives a precise version of the classical Poincaré recurrence theorem for iterates of the boundary extension of an inner function that fixes~0. We generalise this to forward composition sequences $F_n=f_n\circ \dots\circ f_1,$ $n\in \mathbb{N},$ where $f_n$ are inner functions that fix~0, giving conditions on the contraction of $(F_n)$ so that the radial boundary extension $F_n$ hits any shrinking target of arcs $(I_n)$ of a given size. Next, Aaronson, and also Doering and Mañé, gave a remarkable dichotomy for iterates of any inner function, showing that the behaviour of the boundary extension is of two entirely different types, depending on the size of the sequence $(|f^n(0)|)$. In earlier work, we showed that one part of this dichotomy holds in the non-autonomous setting of forward compositions. It turns out that this dichotomy is closely related to the result of Fernández, Melián and Pestana, and here we show that a version of the second part of the dichotomy holds in the non-autonomous setting provided we impose a condition on the contraction of $(F_n)$ in relation to the size of the sequence $(|F_n(0)|)$. The techniques we use include a strong version of the second Borel--Cantelli lemma and strong mixing results of Pommerenke for contracting sequences of inner functions. We give examples to show that the contraction conditions that we need to impose in the non-autonomous setting are best possible.

math.DS↗

arXiv2023

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy--Wolff Theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, for the first time, we address such questions in the very general setting of sequences $(F_n)$ of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. Some of our results are new even in the classical setting. Our methods apply in particular to non-autonomous dynamical systems, when $(F_n)$ are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics. The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a `weak independence' version of the second Borel--Cantelli lemma and the concept from ergodic theory of `shrinking targets'.

math.DS↗

arXiv2021

Iterating the minimum modulus: functions of order half, minimal type

For a transcendental entire function $f$, the property that there exists $r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, where $m(r)=\min \{|f(z)|:|z|=r\}$, is related to conjectures of Eremenko and of Baker, for both of which order $1/2$ minimal type is a significant rate of growth. We show that this property holds for functions of order $1/2$ minimal type if the maximum modulus of $f$ has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg's method of constructing entire functions of small growth, which allows rather precise control of $m(r)$.

math.CV↗

arXiv2020

Oscillating simply connected wandering domains

Although detailed descriptions of the possible types of behaviour inside periodic Fatou components have been known for over 100 years, a classification of wandering domains has only recently been given. Recently, simply connected wandering domains were classified into nine possible types and examples of escaping wandering domains of each of these types were constructed. Here we consider the case of oscillating wandering domains, for which only six of these types are possible. We use a new technique based on approximation theory to construct examples of all six types of oscillating simply connected wandering domains. This requires delicate arguments since oscillating wandering domains return infinitely often to a bounded part of the plane. Our technique is inspired by that used by Eremenko and Lyubich to construct the first example of an oscillating wandering domain, but with considerable refinements which enable us to show that the wandering domains are bounded, to specify the degree of the mappings between wandering domains and to give precise descriptions of the dynamical behaviour of these mappings.

math.DS↗

arXiv2020

On subharmonic and entire functions of small order: after Kjellberg

We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves developing a new technique to give an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, giving a sharp estimate for the growth in many cases.

math.CV↗

arXiv2019

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.

math.DS↗

arXiv2018

Eremenko's conjecture for functions with real zeros: the role of the minimum modulus

We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.

math.DS↗

arXiv2015

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 behaviour 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.

math.DS↗

arXiv2013

Regularity and fast escaping points of entire functions

Let $f$ be a transcendental entire function. The fast escaping set $A(f)$, various regularity conditions on the growth of the maximum modulus of $f$, and also, more recently, the quite fast escaping set $Q(f)$ have all been used to make progress on fundamental questions concerning the iteration of $f$. In this paper we establish new relationships between these three concepts. We prove that a certain weak regularity condition is necessary and sufficient for $Q(f)=A(f)$ and give examples of functions for which $Q(f)\neq A(f)$. We also apply a result of Beurling that relates the size of the minimum modulus of $f$ to the growth of its maximum modulus in order to establish that a stronger regularity condition called log-regularity holds for a large class of functions, in particular for functions in the Eremenko-Lyubich class ${\mathcal B}$.

math.DS↗

arXiv2013

Annular itineraries for entire functions

In order to analyse the way in which the size of the iterates $(f^n(z))$ of a transcendental entire function $f$ can behave, we introduce the concept of the {\it annular itinerary} of a point $z$. This is the sequence of non-negative integers $s_0s_1...$ defined by \[ f^n(z)\in A_{s_n}(R),\;\;\text{for}n\ge 0, \] where $A_0(R)=\{z:|z| 0$ is so large that $M(r)>r$, for $r\ge R$. We consider the different types of annular itineraries that can occur for any transcendental entire function $f$ and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.

math.DS↗

arXiv2011

Baker's conjecture and Eremenko's conjecture for functions with negative real zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.

math.DS↗

arXiv2011

Multiply connected wandering domains of entire functions

The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function $f$ in any multiply connected wandering domain $U$ of $f$. By introducing a certain positive harmonic function $h$ in $U$, related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large $n$, the image domains $U_n=f^n(U)$ contain large annuli, $C_n$, and that the union of these annuli acts as an absorbing set for the iterates of $f$ in $U$. Moreover, $f$ behaves like a monomial within each of these annuli and the orbits of points in $U$ settle in the long term at particular `levels' within the annuli, determined by the function $h$. We also discuss the proximity of $\partial U_n$ and $\partial C_n$ for large $n$, and the connectivity properties of the components of $U_n \setminus \bar{C_n}$. These properties are deduced from new results about the behaviour of an entire function which omits certain values in an annulus.

math.CV↗

arXiv2010

Boundaries of escaping Fatou components

Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $U$ are escaping, then $U$ is an escaping Fatou component. Some applications of these results are given; for example, if $I(f)$ is the escaping set of $f$, then $I(f)\cup\{\infty\}$ is connected.

math.CV↗

arXiv2009

Hausdorff dimensions of escaping sets of transcendental entire functions

Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ ϕ= ψ\circ f$, where $ϕ,ψ:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.

math.DS↗

arXiv2009

Are Devaney hairs fast escaping?

Beginning with Devaney, several authors have studied transcendental entire functions for which every point in the escaping set can be connected to infinity by a curve in the escaping set. Such curves are often called Devaney hairs. We show that, in many cases, every point in such a curve, apart from possibly a finite endpoint of the curve, belongs to the fast escaping set. We also give an example of a Devaney hair which lies in a logarithmic tract of a transcendental entire function and contains no fast escaping points.

math.DS↗

arXiv2008

The growth rate of an entire function and the Hausdorff dimension of its Julia set

Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.

math.CV↗

arXiv2008

Dynamics of meromorphic functions with direct or logarithmic singularities

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.

math.CV↗