On complexification and iteration of quasiregular polynomials which have algebraic degree two
Tom 186 / 2005
Fundamenta Mathematicae 186 (2005), 269-285
MSC: Primary 30C99, 32H50, 37F99; Secondary 30C62, 30D90.
DOI: 10.4064/fm186-3-5
Streszczenie
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.
