I'm studying the paper "Local and simultaneous structural stability of certain diffeomorphisms. - Marco Antonio Teixeira". At the beginning of the paper, the author gives the following definition
Besides that, the author states the following lemma:
However, he does not demonstrate such a result, just saying "the lemma is easy to proof".
I know how to find an extension $\omega:\mathbb{R}^2\to \mathbb{R}^2$ of $\left. \varphi \right|_{U}$ such that $\varphi'(0) + \beta$ where $\beta \in \mathcal C_b^0 (\mathbb{R}^2)$ is Lipschitz with bounded constant by $\varepsilon$. However, I don't have the faintest idea how to guarantee $\omega\circ \omega= \mathrm{Id}.$
Can anyone help me?
How do I constructed the function $\omega$
Define $F= \varphi$ and $L = \mathrm{d} \varphi(0)$.
First, choose a $\mathcal{C}^\infty$ real function $\beta:\mathbb{R} \to \mathbb{R}$ satisfying:
$\beta(x) = 0 $, $\forall$ $x$ $\in$ $\mathbb{R} \setminus [-1,1]$.
$\beta(x) =1$, $\forall$ $x$ $\in$ $ \left[-\frac{1}{2},\frac{1}{2}\right].$
$\beta (x)$ is a increasing function in $\left[-1,-\frac{1}{2}\right]$ and $\beta(x)$ is a decreasing function in $\left[\frac{1}{2}, 1\right]$.
Note that, $\beta$ is compact support $\mathcal{C}^\infty$, so, there exists $M$ $\in$ $\mathbb{R}$, such that, $M = \sup \{|D\beta(x)|; \hspace{0.1cm} x \in \mathbb{R}\} $.
We define $\phi: U \rightarrow \mathbb{R}^2$ as $\phi(x) = F(x) - Lx$. Observe that the function $\phi$ is a $\mathcal{C}^\infty$, moreover, $D\phi(0) = 0$. Therefore, for the real number $\widetilde{\varepsilon} = \min\left\{ \frac{\varepsilon}{2M} , \frac{\varepsilon}{2}\right\} > 0$ there exists $1 \geq r >0$ such that $$x \in B_r (0) \subset U \Rightarrow \Vert D\phi(x) \Vert < \widetilde{\varepsilon} $$ implying, by the mean value theoream that, if $x \in B_r (0) \Rightarrow \Vert \phi(x) \Vert < \widetilde{\varepsilon}\Vert x\Vert $.
Now, consider $\omega: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined as: $$\omega(x) = \left\{\begin{array}{l} Lx + \beta\left(\frac{\Vert x \Vert}{r}\right)\phi(x), \hspace{0.1cm} \text{if} \hspace{0.1cm} x \in U,\\ Lx, \hspace{0.1cm} \text{if} \hspace{0.1cm} x \in \mathbb{R}^n \setminus U. \end{array} \right. $$
Thus, defining $r_1 = r$ and $r_2 = \frac{r}{2}$. It is easy to verify that
- $\omega = F$ in $B_{r_2} (0)$.
- $\omega = L$ outside $B_{r_1}(0)$.
- The function $\alpha = G - L$ is bounded by $\varepsilon$.
- $\alpha$ is $\varepsilon$-Lipschitz.
Once $\omega = L +\alpha$, we constructed such an extension. However, $\omega \circ \omega $ is not necessarily equal to $\mathrm{Id}$.
