On the connectivity of the escaping set in the punctured plane

• Autores: Vasiliki Evdoridou, David Martí-Pere, D. J. Sixsmith
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 1, 2021, págs. 109-127
• Idioma: inglés
• DOI: 10.1007/s13348-020-00282-6
• Enlaces
  • Texto completo
• Resumen

  • We consider the dynamics of transcendental self-maps of the punctured plane, \mathbb {C}^*=\mathbb {C}{\setminus } \{0\}. We prove that the escaping set I(f) is either connected, or has infinitely many components. We also show that I(f)\cup \{0,\infty \} is either connected, or has exactly two components, one containing 0 and the other \infty. This gives a trichotomy regarding the connectivity of the sets I(f) and I(f)\cup \{0,\infty \}, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, we show that Baker domains can be doubly connected in \mathbb {C}^* by constructing the first such example. We also prove that if f has a doubly connected Baker domain, then its closure contains both 0 and \infty, and hence I(f)\cup \{0,\infty \} is connected.

• Referencias bibliográficas
  • Baker, I.N.: Wandering domains in the iteration of entire functions. Proc. Lond. Math. Soc. (3) 49(3), 563–576 (1984)
  • Baker, I.N.: Wandering domains for maps of the punctured plane. Ann. Acad. Sci. Fenn. Ser. A I Math. 12(2), 191–198 (1987)
  • Baker, I.N., Domínguez-Soto, P.: Analytic self-maps of the punctured plane. Complex Var. Theory Appl. 37(1–4), 67–91 (1998)
  • Baker, I.N., Domínguez, P.: Some connectedness properties of Julia sets. Complex Var. Theory Appl. 41(4), 371–389 (2000)
  • Bergweiler, W.: Iteration of meromorphic functions. Bull. Am. Math. Soc. 29(2), 151–188 (1993)
  • Bergweiler, W.: On the Julia set of analytic self-maps of the punctured plane. Analysis 15(3), 251–256 (1995)
  • Barański, K., Fagella, N., Jarque, X., Karpińska, B.: Absorbing sets and Baker domains for holomorphic maps. J. Lond. Math. Soc. (2) 92(1),...
  • Cowen, C.C.: Iteration and the solution of functional equations for functions analytic in the unit disk. Trans. Am. Math. Soc. 265, 69–95...
  • Domínguez-Soto, P.: Connectedness properties of Julia sets of transcendental entire functions. Complex Var. 32, 199–215 (1997)
  • Eremenko, A.E., Lyubich, M.Yu.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42(4), 989–1020 (1992)
  • Evdoridou, V., Martí-Pete, D., Sixsmith, D.J.: Spiders’ webs in the punctured plane. Ann. Acad. Sci. Fenn. Math. 45, 511–531 (2020)
  • Eremenko, A.E.: On the Iteration of Entire Functions, vol. 23, pp. 339–345. Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center...
  • Evdoridou, V.: Fatou’s web. Proc. Am. Math. Soc. 144(12), 5227–5240 (2016)
  • Fagella, N.: Dynamics of the complex standard family. J. Math. Anal. Appl. 229(1), 1–31 (1999)
  • Fagella, N., Martí-Pete, D.: Dynamic rays of bounded-type transcendental self-maps of the punctured plane. Discrete Contin. Dyn. Syst. 37,...
  • Gaier, D.: Lectures on Complex Approximation. Birkhäuser Boston, Inc., Boston (1987). (Translated from the German by Renate McLaughlin)
  • Goldberg, L.R., Keen, L.: A finiteness theorem for a dynamical class of entire functions. Ergod. Theory Dyn. Syst. 6(2), 183–192 (1986)
  • Garnett, J.B., Marshall, D.E.: Harmonic Measure, New Mathematical Monographs, vol. 2. Cambridge University Press, Cambridge (2005)
  • Kisaka, M.: On the connectivity of Julia sets of transcendental entire functions. Ergod. Theory Dyn. Syst. 18(1), 189–205 (1998)
  • Martí-Pete, D.: The escaping set of transcendental self-maps of the punctured plane. Ergod. Theory Dyn. Syst. 38(2), 739–760 (2018)
  • Martí-Pete, D.: Escaping Fatou components of transcendental self-maps of the punctured plane. Math. Proc. Cambridge Philos. Soc. 1–25 (2019)....
  • Nicks, D.A., Sixsmith, D.J.: The dynamics of quasiregular maps of punctured space. Indiana Univ. Math. J. 68(1), 323–352 (2019)
  • Osborne, J.W., Rippon, P.J., Stallard, G.M.: Connectedness properties of the set where the iterates of an entire function are unbounded. Ergod....
  • Osborne, J.W., Sixsmith, D.J.: On the set where the iterates of an entire function are neither escaping nor bounded. Ann. Acad. Sci. Fenn....
  • Osborne, J.W.: Connectedness properties of the set where the iterates of an entire function are bounded. Math. Proc. Cambridge Philos. Soc....
  • Osborne, J.W.: Spiders’ webs and locally connected Julia sets of transcendental entire functions. Ergod. Theory Dyn. Syst. 33(4), 1146–1161...
  • Rådström, H.: On the iteration of analytic functions. Math. Scand. 1, 85–92 (1953)
  • Ransford, T.: Potential Theory in the Complex Plane, London Mathematical Society Student Texts, vol. 28. Cambridge University Press, Cambridge...
  • Rempe, L.: Connected escaping sets of exponential maps. Ann. Acad. Sci. Fenn. Math. 36, 71–80 (2011)
  • Rippon, P.J.: Baker domains, Transcendental Dynamics and Complex Analysis, London Mathematical Society Lecture Note Series, vol. 348, pp....
  • Rippon, P.J., Stallard, G.M.: Boundaries of escaping Fatou components. Proc. Am. Math. Soc. 139(8), 2807–2820 (2011)
  • Rippon, P.J., Stallard, G.M.: Fast escaping points of entire functions. Proc. Lond. Math. Soc. (3) 105(4), 787–820 (2012)
  • Rippon, P.J., Stallard, G.M.: Boundaries of univalent Baker domains. J. Anal. Math. 134(2), 801–810 (2018)
  • Rippon, P.J., Stallard, G.M.: Eremenko points and the structure of the escaping set. Trans. Am. Math. Soc. 372, 3083–3111 (2019)
  • Sixsmith, D.J.: Dynamical sets whose union with infinity is connected. Ergod. Theory Dyn. Syst. 40(3), 789–798 (2020)