Open Research Online: No conditions. Results ordered -Date Deposited. 2024-02-29T16:32:41ZEPrintshttps://oro.open.ac.uk/images/sitelogo.pnghttps://oro.open.ac.uk/2021-12-17T09:57:15Z2023-08-08T12:58:08Zhttps://oro.open.ac.uk/id/eprint/81128This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/811282021-12-17T09:57:15ZOscillating simply connected wandering domainsAlthough 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.Vasiliki Evdoridouve793Philip Ripponpjr9Gwyneth Stallardgms42021-09-02T15:09:32Z2023-05-20T05:16:20Zhttps://oro.open.ac.uk/id/eprint/78693This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/786932021-09-02T15:09:32ZClassifying simply connected wandering domainsWhile 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.Philip Ripponpjr9Gwyneth Stallardgms4Vasiliki Evdoridouve793Nuria FagellaAnna Miriam Benini2021-06-30T09:55:36Z2023-07-01T00:32:58Zhttps://oro.open.ac.uk/id/eprint/77244This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/772442021-06-30T09:55:36ZIterating the minimum modulus: functions of order half, minimal typeFor a transcendental entire function , the property that there exists such that as , where , is related to conjectures of Eremenko and of Baker, for both of which order minimal type is a significant rate of growth. We show that this property holds for functions of order minimal type if the maximum modulus of 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 .Daniel NicksPhilip Ripponpjr9Gwyneth Stallardgms42021-05-05T08:00:21Z2023-07-01T00:21:51Zhttps://oro.open.ac.uk/id/eprint/76142This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/761422021-05-05T08:00:21ZOn Subharmonic and Entire Functions of Small Order: After KjellbergWe 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 a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases.Philip J. Ripponpjr9Gwyneth M. Stallardgms42020-01-27T10:07:35Z2022-04-05T20:44:51Zhttps://oro.open.ac.uk/id/eprint/69153This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/691532020-01-27T10:07:35ZEremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum ModulusWe consider the class of real transcendental entire functions of finite order with only real zeros, and show that if the iterated minimum modulus tends to , then the escaping set of has the structure of a spider's web, in which case Eremenko's conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko's conjecture. For functions in this class we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function .Daniel Nicksdn2627Philip Ripponpjr9Gwyneth Stallardgms42018-06-28T15:18:21Z2021-08-31T21:42:50Zhttps://oro.open.ac.uk/id/eprint/55614This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/556142018-06-28T15:18:21ZEremenko points and the structure of the escaping setMuch recent work on the iterates of a transcendental entire function f has been motivated by Eremenko's conjecture that all the components of the escaping set I(f) are unbounded. We prove several general results about the topological structure of I(f) including the fact that if I(f) is disconnected, then it contains uncountably many pairwise disjoint unbounded continua, all of which are subsets of A_{R}(f), the 'core' of the fast escaping set. We also show that, for some R > 0, the set A_{R}(f) is connected and has the structure of an infinite spider's web or it contains uncountably many unbounded connected F_{σ} sets. There are analogous results for the intersections of these sets with the Julia set when multiply connected wandering domains are not present, but very different results 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.Philip Ripponpjr9Gwyneth Stallardgms42018-02-28T09:50:20Z2021-08-31T15:43:31Zhttps://oro.open.ac.uk/id/eprint/53677This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/536772018-02-28T09:50:20ZBaker's conjecture for functions with real zerosBaker's conjecture states that a transcendental entire functions of order less than 1/2 has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1.

Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1.Daniel A. NicksPhilip J. Ripponpjr9Gwyneth M. Stallardgms42017-03-13T14:39:25Z2021-08-31T15:43:02Zhttps://oro.open.ac.uk/id/eprint/48913This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/489132017-03-13T14:39:25ZThe iterated minimum modulus and conjectures of Baker and EremenkoIn transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape at least as fast as iterates of the minimum modulus, and obtain new results related to Eremenko's conjecture and Baker's conjecture, and the rate of escape in Baker domains. To do this we prove a result of wider interest concerning the existence of points that escape to infinity under the iteration of a positive continuous function.John W. Osbornejo2242Philip J. Ripponpjr9Gwyneth M. Stallardgms42017-03-06T16:26:08Z2023-04-04T05:00:54Zhttps://oro.open.ac.uk/id/eprint/48828This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/488282017-03-06T16:26:08ZThe MacLane class and the Eremenko-Lyubich classIn 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class A to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of C. J. Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class A.Karl F. BarthPhilip J. Ripponpjr9David J. Sixsmithdjs8692015-10-12T10:15:02Z2021-08-31T15:42:47Zhttps://oro.open.ac.uk/id/eprint/44549This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/445492015-10-12T10:15:02ZPermutable entire functions and multiply connected wandering domainsLet 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.Anna Miriam BeniniPhilip J. Ripponpjr9Gwyneth M. Stallardgms42015-08-27T08:44:53Z2021-08-31T15:42:39Zhttps://oro.open.ac.uk/id/eprint/44199This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/441992015-08-27T08:44:53ZConnectedness properties of the set where the iterates of an entire function are unboundedWe investigate the connectedness properties of the set I+(f) of points where the iterates of an entire function f are unbounded. In particular, we show that I+(f) is connected whenever iterates of the minimum modulus of f tend to ∞. For a general transcendental entire function f, we show that I+(f)∪ is always connected and that, if I+(f) is disconnected, then it has uncountably many components, infinitely many of which are unbounded.John Osbornejo2242Philip Ripponpjr9Gwyneth Stallardgms42014-12-02T10:23:37Z2021-08-31T15:43:06Zhttps://oro.open.ac.uk/id/eprint/41450This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/414502014-12-02T10:23:37ZBoundaries of univalent Baker domainsLet be a transcendental entire function and let be a univalent Baker domain of . We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of form a set of harmonic measure zero with respect to . This leads to a new sufficient condition for the escaping set of to be connected, and also a new general result on Eremenko's conjecture.P. J. Ripponpjr9G. M. Stallardgms42014-08-06T10:02:47Z2021-08-31T08:57:43Zhttps://oro.open.ac.uk/id/eprint/40671This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/406712014-08-06T10:02:47ZAnnular itineraries for entire functionsIn order to analyse the way in which the size of the iterates of a transcendental entire function f can behave, we introduce the concept of the annular itinerary of a point z. This is the sequence of non-negative integers
s_{0}s_{1} . . . defined by

f^{n}(z) ∈ A_{s}_{n}(R), for n ≥ 0,

where A_{0}(R) = {z : |z| < R} and

A_{n}(R) = {z : M^{n−1}(R) ≤ |z|<M^{n}(R)}, n≥ 1.

Here M(r) is the maximum modulus of f on {z : |z| = r} and R > 0 is so
large that M(r) > r, for r ≥ 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.P. J. Ripponpjr9G. M. Stallardgms42013-07-11T10:08:56Z2021-08-31T15:42:18Zhttps://oro.open.ac.uk/id/eprint/37989This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/379892013-07-11T10:08:56ZRegularity and fast escaping points of entire functionsLet 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)≠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 ℬ.Philip Jonathan Ripponpjr9Gwyneth Mary Stallardgms42013-07-11T08:22:18Z2021-08-31T15:43:14Zhttps://oro.open.ac.uk/id/eprint/37990This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/379902013-07-11T08:22:18ZA sharp growth condition for a fast escaping spider’s webWe show that the fast escaping set A(f) of a transcendental entire function f has a structure known as a spider’s web whenever the maximum modulus of f grows below a certain rate. The proof uses a new local version of the cos πρ theorem, based on a comparatively unknown result of Beurling. We also give examples of entire functions for which the fast escaping set is not a spider’s web which show that this growth rate is sharp. These are the first examples for which the escaping set has a spider’s web structure but the fast escaping set does not. Our results give new insight into possible approaches to proving a conjecture of Baker, and also a conjecture of Eremenko.P. J. Ripponpjr9G. M. Stallardgms42012-11-02T09:06:54Z2021-08-31T15:42:16Zhttps://oro.open.ac.uk/id/eprint/35313This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/353132012-11-02T09:06:54ZMultiply connected wandering domains of entire functionsThe 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 in any multiply connected wandering domain of . By introducing a certain positive harmonic function in , 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 , the image domains contain large annuli, , and that the union of these annuli acts as an absorbing set for the iterates of in . Moreover, behaves like a monomial within each of these annuli and the orbits of points in settle in the long term at particular `levels' within the annuli, determined by the function . We also discuss the proximity of and for large , and the connectivity properties of the components of . These properties are deduced from new results about the behaviour of entire functions that omit certain values in an annulus.Walter BergweilerPhilip Ripponpjr9Gwyneth Stallardgms42012-01-17T12:01:56Z2021-08-31T08:57:42Zhttps://oro.open.ac.uk/id/eprint/31138This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/311382012-01-17T12:01:56ZFast escaping points of entire functionsLet be a transcendental entire function and let denote the set of points that escape to infinity `as fast as possible’ under iteration. By writing as a countable union of closed sets, called `levels’ of , we obtain a new understanding of the structure of this set. For example, we show that if is a Fatou component in , then and this leads to significant new results and considerable improvements to existing results about . In particular, we study functions for which , and each of its levels, has the structure of an `infinite spider's web’. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.Philip Ripponpjr9Gwyneth Stallardgms42012-01-17T11:53:06Z2021-08-31T08:57:42Zhttps://oro.open.ac.uk/id/eprint/31617This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/316172012-01-17T11:53:06ZBaker's conjecture and Eremenko's conjecture for functions with negative zerosWe 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 ½ 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 fuctions whose zeros all lie on the negative real axis, in particular those of order less than ½. 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.Philip Ripponpjr9Gwyneth Stallardgms42012-01-17T11:51:13Z2021-05-10T10:18:25Zhttps://oro.open.ac.uk/id/eprint/31618This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/316182012-01-17T11:51:13ZExotic Baker and wandering domains for Ahlfors island mapsLet X be a Riemann surface of genus at most 1, i.e. X is the Riemann sphere or a torus. We construct a variety of examples of analytic functions g : W → X, where W is an arbitrary subdomain of X, that satisfy Epstein’s “Ahlfors islands condition”. In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the omega-limit set of a Baker domain of such a function. As a corollary of our construction, we show that there are entire functions with Baker domains in which the iterates converge to infinity arbitrarily slowly. We also construct Ahlfors islands maps with wandering domains and logarithmic singularities, as well as examples where X is a compact hyperbolic surface.Lasse RempePhilip Ripponpjr92012-01-17T11:15:22Z2021-08-31T08:57:42Zhttps://oro.open.ac.uk/id/eprint/30393This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/303932012-01-17T11:15:22ZBoundaries of escaping Fatou componentsLet be a transcendental entire function and be a Fatou component of . We show that if is an escaping wandering domain of , then most boundary points of (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of are escaping, then is an escaping Fatou component. Some applications of these results are given; for example, if is the escaping set of , then is connected.P. J. Ripponpjr9G. M. Stallardgms42012-01-11T15:49:00Z2021-05-10T10:18:24Zhttps://oro.open.ac.uk/id/eprint/23857This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/238572012-01-11T15:49:00ZNon-tangential limits of slowly growing analytic functionsWe show that if is an analytic function in the unit disc, as , for every , and , where then has a finite non-tangential limit at . We also show that in this result it is not sufficient to assume that as , for some fixed .Karl F. BarthPhilip J. Ripponpjr92011-12-08T09:44:49Z2021-08-31T08:57:42Zhttps://oro.open.ac.uk/id/eprint/30391This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/303912011-12-08T09:44:49ZSlow escaping points of meromorphic functionsWe 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.Philip Ripponpjr9Gwyneth Stallardgms42010-09-23T08:46:11Z2021-08-31T08:57:41Zhttps://oro.open.ac.uk/id/eprint/23231This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/232312010-09-23T08:46:11ZAre 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.Lasse RempePhilip J. Ripponpjr9Gwyneth M. Stallardgms42010-07-28T15:22:49Z2021-08-31T08:57:40Zhttps://oro.open.ac.uk/id/eprint/22450This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/224502010-07-28T15:22:49ZOn multiply connected wandering domains of meromorphic functionsWe 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) is contained in V, then f maps each component of the boundary of U onto a component of the boundary of V in C^. 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.P. J. Ripponpjr9G. M. Stallardgms42010-07-28T11:39:44Z2021-08-31T08:57:40Zhttps://oro.open.ac.uk/id/eprint/22425This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/224252010-07-28T11:39:44ZFunctions of small growth with no unbounded Fatou componentsWe prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand, we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen’s condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.P. J. Ripponpjr9G. M. Stallardgms42009-09-08T09:34:41Z2021-08-31T08:57:40Zhttps://oro.open.ac.uk/id/eprint/18351This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/183512009-09-08T09:34:41ZEscaping points of entire functions of small growthLet f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.P. J. Ripponpjr9G. M. Stallardgms42009-05-08T14:46:10Z2021-05-10T10:18:23Zhttps://oro.open.ac.uk/id/eprint/15977This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/159772009-05-08T14:46:10ZExtensions of a theorem of ValironA classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value infinity. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.K. F. BarthPhilip Ripponpjr92009-05-06T14:01:49Z2021-08-31T17:26:13Zhttps://oro.open.ac.uk/id/eprint/16077This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/160772009-05-06T14:01:49ZSingularities of meromorphic functions with Baker domainsP. J. Ripponpjr9G. M. Stallardgms42009-04-14T20:29:28Z2021-08-31T08:57:40Zhttps://oro.open.ac.uk/id/eprint/15753This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/157532009-04-14T20:29:28ZDynamics of meromorphic functions with direct or logarithmic singularitiesLet f be a transcendental meromorphic function and denote by J(f) the Julia set and by I(f) the escaping set. We show that if f has a direct singularity over infinity, then I(f) has an unbounded component and I(f)∩J(f) contains continua. Moreover, under this hypothesis I(f)∩J(f) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f)∩J(f) is 2 and the Hausdorff dimension of J(f) is strictly greater than 1. 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. This method is also applied to complex differential equations.W. BergweilerPhilip J. Ripponpjr9Gwyneth M. Stallardgms42007-05-25Z2021-05-10T10:18:23Zhttps://oro.open.ac.uk/id/eprint/7817This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/78172007-05-25ZBaker domains of meromorphic functionsLet be a transcendental meromorphic function and be an invariant Baker domain of . We obtain a new estimate for the growth of the iterates of in , and we use this estimate to improve an earlier result relating the geometric properties of and the proximity of in to the identity function. We illustrate the latter result by considering transcendental meromorphic functions of the form

where , and , and we show that these functions have Baker domains which contain an unbounded set of critical points and an unbounded set of critical values.P.J. Ripponpjr92007-05-01Z2021-08-31T08:57:40Zhttps://oro.open.ac.uk/id/eprint/7590This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/75902007-05-01ZDimensions of Julia sets of meromorphic functionsWe show that for any meromorphic function the Julia set has constant local upper and lower box dimensions, and , respectively, near all points of with at most two
exceptions. Further, the packing dimension of the Julia set is equal to . Using this result we show that, for any transcendental entire function in the class (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of are equal to 2. Our approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2.P.J. Ripponpjr9G.M. Stallardgms42007-03-09Z2021-08-31T08:57:39Zhttps://oro.open.ac.uk/id/eprint/3846This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/38462007-03-09ZDimensions of Julia sets of meromorphic functions with finitely many polesLet be a transcendental meromorphic function with finitely many poles such that the finite singularities of lie in a bounded set. We show that the Julia set of has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions.P.J. Ripponpjr9G.M. Stallardgms42006-08-03Z2021-09-15T08:16:21Zhttps://oro.open.ac.uk/id/eprint/3628This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/36282006-08-03ZAsymptotic tracts of locally univalent functionsSeveral results are proved which are related to an old problem of G.R. MacLane, namely, whether functions in the class that are locally univalent can have arc tracts. In particular, a proof is given of an assertion of MacLane that if is locally univalent and has no arc tracts, then .Karl F. BarthPhilip J. Ripponpjr92006-07-06Z2023-02-21T03:24:45Zhttps://oro.open.ac.uk/id/eprint/3840This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/38402006-07-06ZOn questions of Fatou and EremenkoLet be a transcendental entire function and let be the set of points whose iterates under tend to infinity. We show that has at least one unbounded component. In the case that has a Baker wandering domain, we show that is a connected unbounded set.P.J. Ripponpjr9G.M. Stallardgms42006-06-30Z2021-05-11T05:27:23Zhttps://oro.open.ac.uk/id/eprint/3625This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/36252006-06-30ZInfinitely many asymptotic values of locally univalent functionsMcMillan and Pommerenke showed that a locally univalent meromorphic function in the disc, without Koebe arcs, has at least three asymptotic values in each boundary arc. The modular function shows that the number three is best possible. We show that if satisfies certain further conditions, each of which narrowly excludes the modular function, then the number of asymptotic values in each boundary arc must be infinite.Karl F. BarthPhilip J. Ripponpjr92006-06-30Z2021-05-10T10:18:22Zhttps://oro.open.ac.uk/id/eprint/3631This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/36312006-06-30ZEven and odd periods in continued fractions of square rootsThe continued fraction for , where is a positive integer, has the periodic form

where is a palindrome and . The period is assumed to be
of minimal length. We give several new results concerning the intriguing question: How can we distinguish between those integers for which is even and
those for which is odd?Philip Ripponpjr9Harold Taylor2006-06-29Z2021-05-10T10:18:22Zhttps://oro.open.ac.uk/id/eprint/3654This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/36542006-06-29ZOn a problem of MacLane concerning arc tractsG.R. MacLane posed the question of whether a locally univalent function in the MacLane class can have an arc tract. We show that the behaviour of any such example must be extremely irregular. We also indicate one possible approach to constructing an example of such a function, which relates MacLane's question to the `type problem' for certain Riemann surfaces.Karl F. BarthPhilip J. Ripponpjr92006-06-29Z2021-05-10T10:18:22Zhttps://oro.open.ac.uk/id/eprint/3655This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/36552006-06-29ZObituary: Irvine Noel Baker 1932-2001An account of the life and work of the complex analyst Professor I.N. Baker, 1932-2001.Philip Ripponpjr92006-06-29Z2021-05-10T10:18:22Zhttps://oro.open.ac.uk/id/eprint/3672This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/36722006-06-29ZExceptional values and the MacLane classKarl BarthPhilip Ripponpjr92006-06-27Z2021-08-31T08:57:39Zhttps://oro.open.ac.uk/id/eprint/3262This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/32622006-06-27ZEscaping points of meromorphic functions with a finite number of polesP. J. Ripponpjr9G. M. Stallardgms42006-06-27Z2021-05-10T10:18:22Zhttps://oro.open.ac.uk/id/eprint/3274This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/32742006-06-27ZAsymptotic values of strongly normal functionsLet f be meromorphic in the open unit disc D and strongly normal; that is,

(1 - |z|^{2}) f^{#} (z) → 0as|z| → 1,

where f^{#} denotes the spherical derivative of f. We prove results about the existence of asymptotic values of f at points of C ∂D. For example, f has asymptotic values at an uncountably dense subset of C, and the asymptotic values of f form a set of positive linear measure.Karl F. BarthPhilip J. Ripponpjr9