I am reading a mathematical text for a group project and one wants to construct a homeomorphism of $\mathbb{R}^n\rightarrow \mathbb{R}^n$ that is fixed on and outside the disc of radius $a$ centered at $0$ ($D_a$) as follows:
May someone please clarify how $\theta_t$ is constructed? I am confused, since $a,b,c,d$ are real numbers and $\theta_t:\mathbb{R}^4\rightarrow Homeo(R^n)$
Pardon the rough drawing, I was too lazy to boot up Ipe. Since the function in question depends only on the distance from the origin, it can be visualized in 1d, as I have done below. A slope of $1$ indicates the identity function, i.e. nothing is moving, these are the black lines on $[0,a]$ and $[d,\infty)$. Next, we choose were to send $b$, in this case $0.5b + 0.5c$, but in general any point between $b$ and $c$ works; this gives the blue dot. Finally, we need to fill in the function on $[a,b]$ and $[b,d]$ -- there are many ways to do this and keep the function continuous, but the simplest is just to make it linear. On $[a,b]$ this is the unique line from $(a,a)$ to $(b,0.5b+0.5c)$, and on $[b,d]$ this is the unique line from $(b,0.5b+0.5c)$ to $(d,d)$.
This map is always the identity on $[a,b] \cup [d,\infty)$. For $t=0$, it sends $b$ to itself, and for $t=1$ it sends $t=c$, and it is extended everywhere else linearly (in the distance from the origin).