All Outputs (13)

Normal Families and Quasiregular Mappings (2023)
Journal Article
Fletcher, A. N., & Nicks, D. A. (2024). Normal Families and Quasiregular Mappings. Proceedings of the Edinburgh Mathematical Society, 67(1), 79-112. https://doi.org/10.1017/s0013091523000640

Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally... Read More about Normal Families and Quasiregular Mappings.

Iterating the Minimum Modulus: Functions of Order Half, Minimal Type (2021)
Journal Article
Nicks, D. A., Rippon, P. J., & Stallard, G. M. (2021). Iterating the Minimum Modulus: Functions of Order Half, Minimal Type. Computational Methods and Function Theory, 21(4), 653–670. https://doi.org/10.1007/s40315-021-00400-w

For a transcendental entire functionf, the property that there exists r> 0 such that mn(r) → ∞ as n→ ∞, 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 signif... Read More about Iterating the Minimum Modulus: Functions of Order Half, Minimal Type.

Which sequences are orbits? (2021)
Journal Article
A. Nicks, D., & J. Sixsmith, D. (2021). Which sequences are orbits?. Analysis and Mathematical Physics, 11(2), Article 53. https://doi.org/10.1007/s13324-021-00493-5

In the study of discrete dynamical systems, we typically start with a function from a space into itself, and ask questions about the properties of sequences of iterates of the function. In this paper we reverse the direction of this study. In particu... Read More about Which sequences are orbits?.

Eremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus (2020)
Journal Article
Nicks, D. A., Rippon, P. J., & Stallard, G. M. (2021). Eremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus. International Mathematics Research Notices, 2021(18), 13946-13974. https://doi.org/10.1093/imrn/rnaa020

We consider the class of real transcendental entire functions f of finite order with only real zeros and show that if the iterated minimum modulus tends to 8, then the escaping set I(f) of f has the structure of a spider's web, in which case Eremenko... Read More about Eremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus.

The dynamics of quasiregular maps of punctured space (2019)
Journal Article
Nicks, D., & Sixsmith, D. J. (2019). The dynamics of quasiregular maps of punctured space. Indiana University Mathematics Journal, 68(1), 323-352. https://doi.org/10.1512/iumj.2019.68.7556

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps... Read More about The dynamics of quasiregular maps of punctured space.

The bungee set in quasiregular dynamics (2018)
Journal Article
Nicks, D. A., & Sixsmith, D. J. (2019). The bungee set in quasiregular dynamics. Bulletin of the London Mathematical Society, 51(1), 120-128. https://doi.org/10.1112/blms.12215

In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is i... Read More about The bungee set in quasiregular dynamics.

Baker's conjecture for functions with real zeros (2018)
Journal Article
Nicks, D. A., Rippon, P., & Stallard, G. (2018). Baker's conjecture for functions with real zeros. Proceedings of the London Mathematical Society, 117(1), 100-124. https://doi.org/10.1112/plms.12124

Baker's conjecture states that a transcendental entire function 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 a... Read More about Baker's conjecture for functions with real zeros.

Periodic domains of quasiregular maps (2017)
Journal Article
Nicks, D. A., & Sixsmith, D. J. (2018). Periodic domains of quasiregular maps. Ergodic Theory and Dynamical Systems, 38(6), 2321-2344. https://doi.org/10.1017/etds.2016.116

We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show th... Read More about Periodic domains of quasiregular maps.

Hollow quasi-Fatou components of quasiregular maps (2016)
Journal Article
Nicks, D. A., & Sixsmith, D. J. (2017). Hollow quasi-Fatou components of quasiregular maps. Mathematical Proceedings, 162(3), https://doi.org/10.1017/S0305004116000840

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in Rd is called hollow if it has a bounded complementary component. We show that for each d?2 there exists a quasiregular ma... Read More about Hollow quasi-Fatou components of quasiregular maps.

The size and topology of quasi-Fatou components of quasiregular maps (2016)
Journal Article
Nicks, D. A., & Sixsmith, D. J. (2017). The size and topology of quasi-Fatou components of quasiregular maps. Proceedings of the American Mathematical Society, 145(2), 749-763. https://doi.org/10.1090/proc/13253

We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the compon... Read More about The size and topology of quasi-Fatou components of quasiregular maps.

Slow escaping points of quasiregular mappings (2016)
Journal Article
Nicks, D. A. (in press). Slow escaping points of quasiregular mappings. Mathematische Zeitschrift, https://doi.org/10.1007/s00209-016-1687-9

This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of... Read More about Slow escaping points of quasiregular mappings.

Superattracting fixed points of quasiregular mappings (2014)
Journal Article
Fletcher, A., & Nicks, D. A. (2016). Superattracting fixed points of quasiregular mappings. Ergodic Theory and Dynamical Systems, 36(3), https://doi.org/10.1017/etds.2014.88

We investigate the rate of convergence of the iterates of an n-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the ima... Read More about Superattracting fixed points of quasiregular mappings.

Foundations for an iteration theory of entire quasiregular maps (2014)
Journal Article
Bergweiler, W., & NICKS, D. (2014). Foundations for an iteration theory of entire quasiregular maps. Israel Journal of Mathematics, 201(1), 147-184. https://doi.org/10.1007/s11856-014-1081-4

The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular m... Read More about Foundations for an iteration theory of entire quasiregular maps.