= Cubic radial coefficient of a quadratic planar map
{title2=$\ell=\operatorname{Re}(\overline\lambda g_{21})$}
In a complex coordinate write $z'=\lambda z+f_{20}z^2+f_{11}z\overline z+f_{02}\overline z^2$, with $|\lambda|=1$. Away from <strong resonances of a planar map>, the quadratic change $z=w+h_{20}w^2+h_{11}w\overline w+h_{02}\overline w^2$ has $h_{20}=f_{20}/(\lambda^2-\lambda)$, $h_{11}=f_{11}/(1-\lambda)$ and $h_{02}=f_{02}/(\overline\lambda^2-\lambda)$. Substituting and collecting $w^2\overline w$ gives $g_{21}=2f_{20}h_{11}+f_{11}h_{20}+f_{11}\overline h_{11}+2f_{02}\overline h_{02}$. The cubic radial coefficient is $\ell=\operatorname{Re}(\overline\lambda g_{21})$; $\ell<0$ gives a supercritical crossing when the multiplier modulus increases through one.
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