# Periodicities of a Complex Dynamical System

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?

• Simul-posted to m.se, math.stackexchange.com/questions/923353/…, without notification to either site. Sep 8 '14 at 7:08
• Sorry. I did not know that one needs to notify before posting into two communities. I am sorry once again. I think you have really referred the same on behalf of me. Mathematically I did not find anything wrong although. However I am more interested in having your answer. Sep 8 '14 at 7:11
• You are not, as a rule, supposed to post the same thing to both websites, period. But, if you do, it is a common courtesy to note what you are doing at both places. This prevents duplication of effort by people who are answering your question, and it helps you by letting each community improve its response based on what happens at the other. Sep 8 '14 at 7:15
• Essentially the same question posted previously by the same user at mathoverflow.net/questions/180278/… Sep 8 '14 at 9:29
• Dr. you are wrong! The other question was about Boundedness and convergence. This question is about Periodicities. Why dont you like to answer instead of putting nontechnical things? Sep 8 '14 at 9:34