Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number.
It is easy to see that $f(\lambda z, \lambda w)=\lambda^{-1}f(z,w)$ for any $\lambda \in \mathbf{C}$. Hence it is a kind of Homogeneous function of degree $-1$.
Now consider a dynamical system $$Z_{n+1}=\frac{\alpha}{z_n}+\frac{\beta}{z_{n-1}}$$
What can say about periodicity of the sequences? Is the above fact about homogeneity useful anyway in deriving periodicities?