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How may I show that for any $p,q\in\mathbb R$, there exist a diffeomorphism $F:\mathbb R\rightarrow\mathbb R$ such that $F(p)=q$ and $F$ to be an identity function outside of a some neighborhood of $p$ ?

My attempts:

By the bump function $\varphi$ such that $\varphi_{\bigl|[-1,1]}=1$ and $\varphi_{\bigl|\mathbb R-(-2,2)}=0$, I defined $g(x):=a\varphi(x)+x$ for a fixed parameter $a$. If I can show $g'(x)>0$ then inverse function theorem and a change of variable solve it.

1- Why $g'(x)>0$?

2- How can I prove it for $\mathbb{R^n}$ ?
bigli
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    What's wrong with $F(x) = x - p + q$? – Michael Albanese Aug 17 '15 at 18:23
  • Sorry, I edit my question. – bigli Aug 17 '15 at 18:26
  • @bigli Is there any restriction on the neighborhood of $p$? If not you can take the $F$ of Michael Albanese and $\mathbb R$ as neighborhood of $p$ – themaker Aug 17 '15 at 18:38
  • non trivial neighborhood. – bigli Aug 17 '15 at 18:41
  • Finding explicit formulas for $F$ is possible, but I've never seen a painless method. If we only want existence, I prefer to define a nice vector field on $\mathbb{R}$, look at the solution of the associated differential equation, and then prove (with the intermediate value theorem) that for some fixed time $t$, the time-$t$ map of the flow sends $p$ to $q$. Then take $F$ to be this time-$t$ map. This works as well on $\mathbb{R}^n$. – D. Thomine Aug 17 '15 at 18:47

1 Answers1

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Sketch: Choose a bump function $g$ with support in $[-1,1]$ such that $g(0)=1.$ For $a>0,$ let $g_a(t) =(q-p)g(t/a).$ If $a$ is large enough, then $\sup_{\mathbb {R}}|g_a'| < 1.$ Consider $x +g_a(x-p)$ for such $a.$

For $\mathbb {R}^n$ we can do something similar. There is no problem defining a real valued bump function $g$ with support in $\{|x|\le 1\}$ such that $g(0)= 1.$ So, changing notation a bit, we look at

$$f_a(x) = x + g((x-p)/a)(q-p).$$

If $a>0$ is large, then we will have $Df_a$ nonsingular everywhere, and $|f_a(y)-f_a(x)|\ge (1/2)|y-x|$ for all $x,y \in \mathbb {R}^n.$ Any such map is a diffeomorphism of $\mathbb {R}^n$ onto $\mathbb {R}^n.$ There's some work left to do, but this is the basic idea.

zhw.
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