Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have $z\mapsto 1-1/z^2$ which has the periodic orbit {$1,0,\infty$} on the Riemann sphere. Next, I want to calculate the corresponding multiplier $\lambda= (f^{\circ n})' (z_i)=f'(z_1)\cdots f'(z_n)$. In this example we have $f'(z)=2/z^3$, hence $\lambda=f'(1) f'(0) f'(\infty)=2/1 \cdot 2/0 \cdot 2/\infty =\, ???$.

What is wrong here? (I shall prove that $\lambda=0$).

Thanks for your help!

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

Computing the derivative of $f^{\circ 3}(z)$ yield $$-\frac{8 z^3 \left(z^2-1\right)^3}{\left(2 z^2-1\right)^3}$$

Setting $z=1$ yields 0 in the numerator, and 1 in the denominator. Hence, $\lambda = 0.$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.