I am trying to construct an automorphism $$\phi:\mathbb{\overline B}^n\to\mathbb{\overline B}^n$$ such that $\phi(0) = \alpha\hat x_1$, and $\phi|_{\partial\mathbb{\overline B}^n} =$ Id.
I thought about trying to generalize the automorphisms of the unit disk in $\mathbb{C}$, but such automorphims are analytic, which implies if the function is the identity on the boundary, then it is the identity everywhere.
So, I have been playing around with constructions of homeomorphisms, and intuitively I know what such a function should look like, but I can't explicitly constuct it.
I thank you all in advance for your help.
There are many ways to construct such automorphisms; here's one. Let $B$ denote the closed unit ball, and fix a point $p$ in the interior of $B$; we want to find a homeomorphism $\phi:B\to B$ which fixes $\partial B$ and sends $0$ to $p$. Define $\phi$ as follows. Given a point $t\alpha\in B$, with $\|\alpha\|=1$ and $t\in [0,1]$, define $\phi(t\alpha)=t\alpha+(1-t)p$. Geometrically, this can be described as sending each line segment from $0$ to a point of $\partial B$ linearly to the line segment from $p$ to the same point of $\partial B$. It is not hard to show that this $\phi$ is a homeomorphism, and it clearly fixes $\partial B$ and sends $0$ to $p$.