A question concerning orientation preserving interval diffeomorphisms

Question

I'm trying to prove the following:

Let $f,g:I\to I$ , where $I=[a,b]\subset\mathbb{R}$ is some compact interval, be $C^1$ orientation preserving interval diffeomorphisms each with exactly two fixed points, then $f, g$ are topologically conjugate

$\textbf{My attempt:}$ I know I need to show that both maps have exactly one attracting fixed point and one repelling fixed point. I was able to show that it has two fixed points using the intermediate value theorem and the definition of continuity, but I'm lost on how to show that one is attracting and the other repelling. After that I would need to construct fundamental domains but I'm still unsure how to do that and piece it all together.

@Glougloubarbaki some arbitrary compact interval on the real line — user119264, Feb 12 '18 at 10:31

Arnaud Mortier · Answer 1 · 2018-02-12T12:51:36.823

2

Hint:

Consider the graph of the diffeomorphism as a part of a square $[a,b]^2$. The curve has to lie completely above the main diagonal, or completely below, except for the two endpoints which correspond to the fixed points $a$, $b$. In the first case (curve above the diagonal), $b$ is the attractive one, whereas in the second case $a$ is the attractive one.

Hint 2:

Take any point $x$ in your interval, that is not $a$ or $b$. Then the interval $[x, f(x))$ is a fundamental domain (in case $f(x)>x$; otherwise take $[f(x), x)$ instead).

edited Feb 12 '18 at 12:51

answered Feb 12 '18 at 10:29

Arnaud Mortier

27,276

Writing this down mathematically, I'm thinking I should use the mean value theorem then exploit one of the properties of the diffeomorphism to proceed? – user119264 Feb 12 '18 at 10:37
@user119264 I'm not sure exactly what you mean here. Where are you stuck? – Arnaud Mortier Feb 12 '18 at 10:51
I wish to write a mathematical proof, or write what you say formally, but while I agree what you say is certainly true, I can't find a way to write it down mathematically. – user119264 Feb 12 '18 at 10:52
I see. Is it clear that $\phi(a)=a$ and same for $b$? That's because $\phi$ respects the orientation (in other words, is increasing) and is bijective. Now the position wrt the diagonal is because there are no other fixed points. Finally, you can see which fixed point is attractive by considering that in the first case, $f(x)>x$ for all $x\neq a, b$. – Arnaud Mortier Feb 12 '18 at 10:57
how would one go about finding the fundamental domain? – user119264 Feb 12 '18 at 12:42
@user119264 I'll add a hint about that. – Arnaud Mortier Feb 12 '18 at 12:47
Thanks! That was a really useful hint! – user119264 Feb 12 '18 at 12:51
@user119264 you're welcome :) – Arnaud Mortier Feb 12 '18 at 12:52

A question concerning orientation preserving interval diffeomorphisms

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