What's the precise definition of self-similarity of the Cantor set $C$?
It seems we have left and right transformations which each transform the set onto congruent proper subsets of itself whose union is the set itself:
$$T_L(x)=\frac{x}{3}$$
$$T_R(x)=\frac{x+2}{3}$$
Now Wikipedia says $T_L(C)\cong T_R(C)\cong C$ which is true, but but I'm not clear if congruence is a sufficiently precise relation and if it's perhaps a mistake to state such a trivial relation because if I understand congruence correctly, $T_L(X)\cong T_R(X)\cong X$ for pretty much any set of points $X$ on the real line.
What IS unique about the Cantor set is that $T_L(C)\cong T_R(C)\cong T_L(C)\cup T_R(C)$
And the most precise statement it seems to me is:
$$C=T_L(C)\cup T_R(C)$$
A set $A\subset{\mathbb R}^d$ is self similar if $$A=\bigcup_{k=1}^n A_k\ ,$$ whereby $A_k=T_k(A)$ for similarities $T_k:\>{\mathbb R}^d\to{\mathbb R}^d$ $(1\leq k\leq n)$, and the $A_k$ are "almost disjoint" (The exact definition is quite technical. Intuitively the intersections $A_j\cap A_k$ have to be less fat than the $A_k$ themselves.) In the relevant cases the $T_k$ all have scaling factors $\lambda_k<1$.
For the Cantor set we have $d=1$, $n=2$, and scaling factors $\lambda_k={1\over3}$.