$\mathbb{R}\to\mathbb{R}$ Diffeomorphism with special property

Question

$\phi:\mathbb{R}\to\mathbb{R}$ be a diffeomorphism with the following property

$a\in\mathbb{R},|a|<\frac{1}{10}$

(i)$\phi(a)=0$

(ii) $\phi(x)=x$ when $|x|>1$

and how to generalize that in $\mathbb{R}^n$

please help.

Answer 1

First let's generalize: Suppose $a\in B_{\frac{1}{10}}(0)$; the idea is to construct a vector field in $\mathbb{R}^n$, such that in a neighbourhood which contains $0$, $a$ and is contained in $B_1(0)$, the field points in the "direction" of the vector $-a$ with positive velocity, and in the complement, the field is zero.

If we construct this field in such a way that it is Lipschitz, then we know that there exist a flow associated to the field, i.e. if the field is given by $x'(t)=f(x(t))$ with $x(0)=a$, where $f$ is Lipschitz, then there exist a function $F$ depending on $t$ and $a$, satisfying $$\frac{\partial F(a,t)}{\partial t}=f(F(a,t))$$

Moreover, this functions has some good properties:

I- It is $C^1$,

II- $F(a,0)=a$,

III- $F(a,s+t)=F(F(a,t),s)$

IV- For each fixed $t$, $F(a,t):\mathbb{R}^n\rightarrow\mathbb{R}^n$ is an diffeomorphism

Case 1: $\mathbb{R}$

Let $\epsilon>0$ be a small number and construct an $C^1$ function $g:\mathbb{R}\rightarrow [0,1]$, such that $g(t)=1$ if $t\in [0,a]$; $0\leq g(t)\leq 1$ if $t\in (-\epsilon,0)\cup (a,a+\epsilon)$ and $g(t)=0$ on the rest. Consider the vector field $$x'(t)=-g(t),\ x(0)=b$$

Let $F(b,t)$ be the flow associated with this vector field. Note that for $b\in [0,a]$ and $t\in [0,b]$ your flow is defined by $F(b,t)=b-t$. This implies that if $b=a$, then $F(a,a)=0$. Because $F(b,a)$ (b varying) is an diffeomorphism and $F(b,a)=b$ for $b$ in the set where $g$ is zero, you can conclude.

Case 2: $\mathbb{R}^n$

Try to do this case.