Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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A CHARACTERIZATION OF MANDELBROT SET OF QUADRATIC RATIONAL MAPS

  • AHN, YOUNG JOON (Department of Mathematics Education College of Education Chosun University)
  • Received : 2005.03.08
  • Published : 2005.09.25
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Abstract

We present some properties characterizing the Mandelbrot set of quadratic rational maps. Any quadratic rational map is conjugate to either $z^2+c$ or ${\lambda}(z+1/z)+b$. For ${\mid}{\lambda}{\mid}=1$, we find the figure of the Mandelbrot set $M_{\lambda}$, the set of parameters b for which the Julia set of ${\lambda}(z+1/z)+b$ is connected. It is seen to be the whole complex plane if ${\lambda}{\neq}1$, but it is intricate fractal if ${\lambda}=1$. This supplements the work already investigated for the case ${\mid}{\lambda}{\mid}>1$.

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References

  1. Iteration of Rational Functions Beardon, A.F.
  2. An introduction to chaotic Dynamical systems Devaney, R.L.
  3. C. R. Acad. Sci. Paris v.294 Iteration des polynomes quadratiques complexes Douady, A.;Hub-bard, J.
  4. Bull. Soc. Math. France v.47 Sur les equations fonctionnelles Fatou, P.
  5. invent. Math. v.101 The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift Goldberg, L.;Keen, L.
  6. Experimental Mathematics v.2 Geometry and dynamics of quadratic rational maps Milnor, J.
  7. The Beauty of Fractals Peitgen, H.O.;Richter, P.H.
  8. Ann. Math., 2nd Ser. v.43 no.4 Iteration of analytic functions Siegel, C.L.
  9. Letures on Celestial mechanics Siegel, C.L.;Moser, J.
  10. Ann. Sci. Ecole Norm. Sup. On the quasiconformal surgery of rational functions Shishikura, M.
  11. Ann. Math. Conformal homeomorphisms and dynamics I : Solution of the Fatou-Julia problem on wandering domains Sullivan, D.
  12. Complex Variables v.18 On the Julia sets of quadratic rational maps Yin, Y.