= Mandelbrot set
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{title2=$\mathcal M$}
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For $f_c(z)=z^2+c$, the Mandelbrot set is $\mathcal M=\{c:\{f_c^n(0)\}_{n\geq0}\text{ is bounded}\}$. Thus it records the nonescaping finite critical orbit. The <connected Julia set criterion for a polynomial> says that the corresponding <filled Julia set> is connected exactly for these quadratic parameters. Parameters with an attracting fixed point form the main cardioid: its multiplier $\lambda$ gives the fixed point $\lambda/2$ and parameter $c=\lambda/2-\lambda^2/4$, with $|\lambda|<1$. Its boundary has a cusp at $c=1/4$. There the critical orbit increases along the real interval to the parabolic fixed point $1/2$, so the cusp belongs to the set. The real section extends from $-2$ to $1/4$; the familiar period-two bulb is attached at $-3/4$.
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