In most of paper about iterated function systems and fractals which I read, an iterated function system is taken as finite set of contraction mappings like $\{f_1,f_2,...,f_n\}$. In some paper, the number of contraction mappings is taken as $n\geq2$.
Why is there a restriction about the number of contraction mappings in some paper?
The restriction $n \ge 2$ excludes a trivial case. If $n=1$ then there is only a single contraction mapping, and then the fixed set
$$S = \bigcup_{i=1}^n f_i(S) = f_1(S)$$
is simply the (unique) fixed point of $f_1$, which is not very interesting.