On complexification and iteration of quasiregular polynomials which have algebraic degree two

Volume 186 / 2005

Ewa Ligocka Fundamenta Mathematicae 186 (2005), 269-285 MSC: Primary 30C99, 32H50, 37F99; Secondary 30C62, 30D90. DOI: 10.4064/fm186-3-5


We prove that each degree two quasiregular polynomial is conjugate to $Q(z)=z^{2}-(p+q)|z|^{2}+pq\overline{z}^{2}+c$, $|p|<1$, $|q|<1$. We also show that the complexification of $Q$ can be extended to a polynomial endomorphism of $\mathbb{C}\mathbb{P}^{2}$ which acts as a Blaschke product $\frac{z-p}{1-\overline{p}z}\cdot \frac{z-q}{1-\overline{q}z}$ on $\mathbb{C}\mathbb{P}^{2}\setminus\mathbb{C}^{2}$. Using this fact we study the dynamics of $Q$ under iteration.


  • Ewa LigockaInstitute of Mathematics
    Department of Mathematics, Computer Science and Mechanics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland

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