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Mandelbrot and Julia Set

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Sk. Sarif Hassan
Sk. Sarif Hassan 2015-2-4
Consider a dynamical system $$z_{n+1}=\frac{\alpha+z_n}{1+z_{n-1}}$$ for $n=0,1,2,\dots$
In other words the system is $$z_{n+1}=f_{\alpha}(z_n,z_{n-1})$$ where $f_{\alpha}$ is defined from $B(z,r)\times B(z,r)$ to $B(z,r)$ as $f_{\alpha}(z,w)=\frac{\alpha+z}{1+w}$. $B(z,r)$ is an open ball in complex plane.
How can we find out the Mandelbrot and Julia sets for this system?

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