Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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BLASCHKE PRODUCTS AND RATIONAL FUNCTIONS WITH SIEGEL DISKS

  • Katagata, Koh (INTERDISCIPLINARY GRADUATE SCHOOL OF SCIENCE AND ENGINEERING SHIMANE UNIVERSITY)
  • Published : 2009.01.31
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Abstract

Let m be a positive integer. We show that for any given real number ${\alpha}\;{\in}\;[0,\;1]$ and complex number $\mu$ with $|\mu|{\leq}1$ which satisfy $e^{2{\pi}i{\alpha}}{\mu}^m\;{\neq}\;1$, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number $\alpha$. Moreover if the given real number $\alpha$ is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.

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References

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Cited by

  1. DYNAMICS OF TRANSCENDENTAL ENTIRE FUNCTIONS WITH SIEGEL DISKS AND ITS APPLICATIONS vol.48, pp.4, 2011, https://doi.org/10.4134/BKMS.2011.48.4.713