Question:
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If $f(x_0)=x_0,f'(x_0)=1$ and $f''(x_0)>0$, is $x_0$ weakly attracting, weakly repelling, or neither? (weakly attracting meaning $\exists\delta,\forall x\in B_\delta(x_0)\setminus\{x_0\},\forall n: f^n(x)\in B_\delta(x_0)$, and weakly repelling meaning not weakly attracting).
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Repeat 5, but this time assume that $f''(x_0)<0$.
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If $f(x_0)=x_0,f'(x_0)=1,f''(x_0)=0$ and $f'''(x_0)>0$, then show that $x_0$ is weakly repelling.
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Repeat 7, but this time assume $f'''(x_0)<0$ and show that $x_0$ is weakly attracting.
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Combine the results of 5-8 to state a Neutral Fixed Point Theorem.
Motivation: I am taking a course in dynamical systems and we are using Devaney's A First Course in Dynamical Systems. The series of calculus questions presented above are from p. 51 of the book (though I have paraphrased them). The Neutral Fixed Point Theorem is opposed to Attracting/ Repelling Fixed Point Theorems, which say that if $|f'(x_0)|<1$/ $|f'(x_0)|>1$, then $x_0$ is an attracting/ repelling fixed point (the definitions of these are similar, "weakly" denotes only the slowness in the above case). I can easily answer these questions geometrically (the answers for 5 and 6 are that $x_0$ is neither w.a. nor w.r., it repels from one side and attracts from the other, e.g., $f(x)=x-x^2$) but I am looking for analytical proofs.
I have been thinking about these for a while, but all I could come up with are MVT-esque expressions for points $x$ close enough to $x_0$.
P.S.: These are not homework questions.
Best Answer
Here is the key idea for your example $f(x)=x-x^2$ for which the origin is a weakly attractive fixed point on the positive side. Since $0<f(x)<x$ for every $x\in(0,1)$, the orbit of any initial seed $x_1\in(0,1)$ leads to a decreasing sequence which is bounded below by zero. The sequence must, therefore, be convergent and it must converge to the only available fixed point, namely zero. I think it's easier to see that zero is repelling on the negative side.
A similar argument works on the positive side if $f(x)=x-ax^n+O(x^{n+1}))$ for integer $n>1$ and $a>0$.
If $f(x)=-x+ax^n$, you can study $f^2(x)=x-na^2 x^{2n-1}+O(x^{2n})$.
Polynomials with other neutral fixed points are similar. In fact, there is a local conjugacy that allows the results to be extended immediately.