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I have the next function: $f(x) = (1+x^{2N})^{\frac{1}{2Np}}$, where $T_f (x)= (1-p)x - \frac{p}{x^{2N-1}}$, I want to show that $\exists p \in (0,1), N \in \mathbb{N}$ s.t $T^2_f(x)$ has an attractor.

I am given as hint to find for $x \in \mathbb{R}$ s.t $|(T^{2})'(x)|<1$, but I am finding it quite hard to find such an $x$.

I assume this is well know problem, so if you have a reference where I can read on this problem it will help me, or if you wish to solve it or give me greater hints, anyway is good for me.

Thanks in advance.

MathematicalPhysicist
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    Could you clarify a couple if things: 1. Is it a homework? 2. What is the relation between functions $f$ and $T_f$, since f does not appear in the question itself. – Moishe Kohan Feb 03 '14 at 10:34
  • Yes, it's homework. $T_f$ is defined as $ T_f (x) = x - \frac{f(x)}{f'(x)}$. Thanks in advance. – MathematicalPhysicist Feb 03 '14 at 12:34
  • Iterating $T_f$ corresponds to implementing a Newton's method. This problem is different, though, because $f$ has no roots on the real line. The hint instead suggests the existence of a period two sink for $T_f$. – A Blumenthal Feb 07 '14 at 16:00

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