Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, with the upper and lower bounds of the norm of derivative of $f$: $$ \|df(x)\| \leq \alpha < 1, \|df^{-1}(x)\| \leq \beta, ~\forall x\in \mathcal{B} $$ where $\alpha$ and $\beta$ are real constants.

Could $f$ be extended to a $C^k$ function $F$ from the whole space $\mathbb{R}^n$ to itself, still satisfying $$ \|dF(x)\| \leq \alpha, \|dF^{-1}(x)\| \leq \beta, ~\forall x\in \mathbb{R}^n $$ and additionally $$ F(x) \rightarrow \infty \mathrm{~when~} x \rightarrow \infty ? $$

Hence we can use the Hadamard's global inverse function theorem to show that $F$ is a diffeomorphism.

There are related results such as Kirszbraun’s theorem and Whitney's extension theorem, but they are not enough.

If it is true, this result will lead to the global stable and unstable manifold theorem in the thoeory of dynamical systems.

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower bounds of the norm of derivative of $f$: $$ \|df(x)\| \leq \alpha < 1, \|df^{-1}(x)\| \leq \beta, ~\forall x\in \mathcal{B} $$ where $\alpha$ and $\beta$ are real constants.

Could $f$ be extended to a $C^k$ function $F$ from the whole space $\mathbb{R}^n$ to itself, still satisfying $$ \|dF(x)\| \leq \alpha, \|dF^{-1}(x)\| \leq \beta, ~\forall x\in \mathbb{R}^n $$ and additionally $$ F(x) \rightarrow \infty \mathrm{~when~} x \rightarrow \infty ? $$

Hence we can use the Hadamard's global inverse function theorem to show that $F$ is a diffeomorphism.

There are related results such as Kirszbraun’s theorem and Whitney's extension theorem, but they are not enough.

If it is true, this result will lead to the global stable and unstable manifold theorem in the thoeory of dynamical systems.

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, with the upper and lower bounds of the norm of derivative of $f$: $$ 0 < \beta \leq |df(x)| \leq \alpha < 1 ~\forall x\in \mathcal{B} $$ where $\alpha$ and $\beta$ are real constants.

Could $f$ be extended to a $C^k$ function $F$ from the whole space $\mathbb{R}^n$ to itself, still satisfying $$ \beta \leq |dF(x)| \leq \alpha ~\forall x\in \mathbb{R}^n $$ and additionally  $$ F(x) \rightarrow \infty \mathrm{~when~} x \rightarrow \infty ? $$

Hence we can use the Hadamard's global inverse function theorem to show that $F$ is a diffeomorphism.

There are related results such as Kirszbraun’s theorem and Whitney's extension theorem, but they are not enough.

If it is true, this result will lead to the global stable and unstable manifold theorem in the thoeory of dynamical systems.

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, with the upper and lower bounds of the norm of derivative of $f$: $$ \|df(x)\| \leq \alpha < 1, \|df^{-1}(x)\| \leq \beta, ~\forall x\in \mathcal{B} $$ where $\alpha$ and $\beta$ are real constants.

Could $f$ be extended to a $C^k$ function $F$ from the whole space $\mathbb{R}^n$ to itself, still satisfying $$ \|dF(x)\| \leq \alpha, \|dF^{-1}(x)\| \leq \beta, ~\forall x\in \mathbb{R}^n $$ and additionally  $$ F(x) \rightarrow \infty \mathrm{~when~} x \rightarrow \infty ? $$

Hence we can use the Hadamard's global inverse function theorem to show that $F$ is a diffeomorphism.

There are related results such as Kirszbraun’s theorem and Whitney's extension theorem, but they are not enough.

If it is true, this result will lead to the global stable and unstable manifold theorem in the thoeory of dynamical systems.