Rotation Domains and Stable Baker Omitted Value

• Ghora, Subhasis ^[1] ; Nayak, Tarakanta ^[1]
  1. [1] Indian Institute of Technology, Bhubaneswar
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 3, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00527-0
• Enlaces
  • Texto completo
• Resumen

  • A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of f−1(D) is bounded. Assuming all the iterates fn to be analytic in a neighborhood of its bov, this article proves a number of results on the rotation domains of the function. The number of p-periodic Herman rings is shown to be finite for each p≥1. It is also proved that every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. As a corollary we have shown that if D is a p-periodic rotation domain of f and fp is univalent in a neighbourhood of the boundary ∂D of D then f has no repelling p-periodic point on ∂D. Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be a singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of fn on W is either a parabolic periodic point or in the ω-limit set of recurrent critical points. Using the same ideas, the boundaries of rotation domains are shown to be in the ω-limit set of recurrent critical points.

• Referencias bibliográficas
  • 1. Baranski, K., Fagella, N., Jarque, X., Karpinska, B.: Connectivity of Julia sets of Newton maps: a unified approach. Rev. Mat. Iberoam...
  • 2. Beardon, A. F.: Iteration of rational functions. Complex analytic dynamical systems, Springer-Verlag, New York, (1991)
  • 3. Bergweiler, W.: Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S) 29(2), 151–188 (1993)
  • 4. Bergweiler, W., Terglane, N.: Weakly repelling fixed points and the connectivity of wandering domains. Trans. Amer. Math. Soc. 348(1),...
  • 5. Chakra, T.K., Chakraborty, G., Nayak, T.: Baker omitted value. Complex Var. Elliptic Equ. 61(10), 1353–1361 (2016)
  • 6. Chakra, T. K., Chakraborty, G., Nayak, T.: Herman rings with small periods and omitted values. Acta Math. Sci. Ser. B (Engl. Ed.), 38(6),...
  • 7. Fagella, N., Jarque, X., Taixés, J.: On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points...
  • 8. Ghora, S., Nayak, T.: On periods of Herman rings and relevant poles. Indian J. Pure Appl. Math. (2021). https://doi.org/10.1007/s13226-021-00112-w
  • 9. Ghora, S., Nayak, T., Sahoo, S.: On Fatou sets containing Baker omitted value. J. Dyn. Diff. Equat. (2021). https://doi.org/10.1007/s10884-021-10060-y
  • 10. Nayak, T.: On Fatou components and omitted values, Geometry, groups and dynamics. Contemp. Math., Amer. Math. Soc., Providence, RI, 639,...
  • 11. Nayak, T., Zheng, J.H.: Omitted values and dynamics of transcendental meromorphic functions. J. Lond. Math. Soc. 83(1), 121–136 (2011)
  • 12. Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoamericana...
  • 13. Shishikura, M., Tan, L.: An alternative proof of Mane’s theorem on non-expanding Julia sets. Mandelbrot Set, Theme Variat 274(1), 265–279...
  • 14. Guizhen, C., Wenjuan, P., Lei, T.: On the topology of wandering Julia components. Discrete Contin. Dyn. Syst.-A 29, 929–952 (2011)
  • 15. Imada, M.: Periodic points on the boundaries of rotation domains of some rational functions. Osaka J. Math. 51(1), 215–225 (2014)
  • 16. Nayak, T.: Herman rings of meromorphic maps with an omitted value. Proc. Amer. Math. Soc. 144, 587–597 (2016)
  • 17. Rempe-Gillen, L., Sixsmith, D.: On connected preimages of simply-connected domains under entire functions. Geom. Funct. Anal. 29(5), 1579–1615...
  • 18. Zheng, J.H.: Singularities and limit functions in iteration of meromorphic functions. J. Lond. Math. Soc. 67(2), 195–207 (2003)
  • 19. Zheng, J.H.: Remarks on Herman rings of transcendental meromorphic functions. Indian J. Pure Appl. Math. 31(7), 747–751 (2000)