= Main cardioid of the Mandelbrot set
{title2=$c=\lambda/2-\lambda^2/4,\quad |\lambda|<1$}
= Main cardioid
{synonym}
The main cardioid consists of quadratic parameters with an attracting fixed point. If that fixed point has multiplier $\lambda$, then it is $\lambda/2$ and $c=\lambda/2-\lambda^2/4$, $|\lambda|<1$. The map is injective on the open unit disc: equality of two parameter values forces either equal multipliers or their sum to be two, impossible for distinct interior multipliers. Its boundary is the cardioid obtained from $\lambda=e^{it}$. At $\lambda=1$ the parameter derivative vanishes, giving the cusp $c=1/4$. Parameters just to the real right of that cusp have monotonically escaping critical orbits, while the cusp's critical orbit tends to $1/2$, so it is on the <Mandelbrot set> boundary.
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