Illinois Journal of Mathematics

Fixed curves near fixed points

Alastair Fletcher

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $H$ be a composition of an $\mathbb{R}$-linear planar mapping and $z\mapsto z^{n}$. We classify the dynamics of $H$ in terms of the parameters of the $\mathbb{R}$-linear mapping and the degree by associating a certain finite Blaschke product. We apply this classification to this situation where $z_{0}$ is a fixed point of a planar quasiregular mapping with constant complex dilatation in a neighbourhood of $z_{0}$. In particular, we find how many curves there are that are fixed by $f$ and that land at $z_{0}$.

Article information

Illinois J. Math., Volume 59, Number 1 (2015), 189-217.

Received: 27 April 2015
Revised: 18 November 2015
First available in Project Euclid: 11 February 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]


Fletcher, Alastair. Fixed curves near fixed points. Illinois J. Math. 59 (2015), no. 1, 189--217. doi:10.1215/ijm/1455203164.

Export citation


  • A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications, Narosa, New Delhi, 2007.
  • W. Bergweiler, Iteration of quasiregular mappings, Comput. Methods Funct. Theory 10 (2010), 455–481.
  • W. Bergweiler, Fatou–Julia theory for non-uniformly quasiregular maps, Ergodic Theory Dynam. Systems 33 (2013), 1–23.
  • B. Bielefeld, S. Sutherland, F. Tangerman and J. J. P. Veerman, Dynamics of certain nonconformal degree-two maps of the plane, Experiment. Math. 2 (1993), no. 4, 281–300.
  • B. Bozyk and B. B. Peckham, Dynamics of nonholomorphic singular continuations: A case in radial symmetry, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23 (2013), no. 11, Art. ID 1330036.
  • B. Branner and N. Fagella, Quasiconformal surgery in holomorphic dynamics, Cambridge University Press, Cambridge, 2014.
  • H. Bruin and M. van Noort, Nonconformal perturbations of $z\mapsto z^2+c$: The $1:3$ resonance, Nonlinearity 17 (2004), no. 3, 765–789.
  • C. Cao, A. Fletcher and Z. Ye, Epicycloids and Blaschke products, available at \arxivurlarXiv:1504.06539.
  • M. D. Contreras, S. Diaz-Madrigal and C. Pommerenke, Iteration in the unit disk: The parabolic zoo, Complex and harmonic analysis, DEStech Publ., Inc., Lancaster, PA, 2007, pp. 63–91.
  • A. Fletcher, Unicritical Blaschke products and domains of ellipticity, Qual. Theory Dyn. Syst. 14 (2015), no. 1, 25–38.
  • A. Fletcher and R. Fryer, On Böttcher coordinates and quasiregular maps, Contemp. Math. 575 (2012), 53–76.
  • A. Fletcher and R. Fryer, Dynamics of mappings with constant complex dilatation, to appear in Ergodic Theory Dynam. Systems.
  • A. Fletcher and D. Goodman, Quasiregular mappings of polynomial type in $\R^2$, Conform. Geom. Dyn. 14 (2010), 322–336.
  • A. Fletcher and V. Markovic, Quasiconformal mappings and Teichmüller spaces, Oxford University Press, Oxford, 2007.
  • A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 1, 205–222.
  • T. Iwaniec and G. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 241–254.
  • Y. Jiang, Asymptotically conformal fixed points and holomorphic motions, Ann. Acad. Sci. Fenn. Math. 34 (2009), 27–46.
  • M. Mandell and A. Magnus, On convergence of sequences of linear fractional transformations, Math. Z. 115 (1970), 11–17.
  • J. Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006.
  • B. B. Peckham, Real perturbation of complex analytic families: Points to regions, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 8 (1998), no. 1, 73–93.
  • B. B. Peckham and J. Montaldi, Real continuation from the complex quadratic family: Fixed-point bifurcation sets, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 (2000), no. 2, 391–414.
  • S. Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 26, Springer, Berlin, 1993.
  • D. Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725–752.