We call a set
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
on the real line
![\mathbb{R}](/media/m/1/4/0/140a3cd0f5aa77f0f229f3ae2e64c0a6.png)
superinvariant if for any stretching
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
of the set by the transformation taking
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
to
![A(x) = x_0 + a(x - x_0), a > 0](/media/m/2/e/e/2ee2be0a47eb4b0a46094235f555c45e.png)
there exists a translation
![B(x) = x+b,](/media/m/6/7/f/67f61f66293f76f96b962ecb378ea79a.png)
such that the images of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
under
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
agree; i.e., for any
![x \in S](/media/m/4/e/7/4e78c89088ce0d85fec01c71b592100a.png)
there is a
![y \in S](/media/m/9/5/2/952c564be0c385be20e4e3637daadf1e.png)
such that
![A(x) = B(y)](/media/m/b/d/e/bde0b8a5acbd38fbb7aa11873d5c2121.png)
and for any
![t \in S](/media/m/0/9/4/094fa56b5baaa603ad90e6f569d55c25.png)
there is a
![u \in S](/media/m/4/2/f/42faba7e03b794da5e113c24b7d81ad6.png)
such that
![B(t) = A(u).](/media/m/f/e/3/fe35d3fa679e839c4e6f28a3f7f3687e.png)
Determine all superinvariant sets.
%V0
We call a set $S$ on the real line $\mathbb{R}$ superinvariant if for any stretching $A$ of the set by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0), a > 0$ there exists a translation $B,$ $B(x) = x+b,$ such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$ there is a $y \in S$ such that $A(x) = B(y)$ and for any $t \in S$ there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.