Addresses of IFS

The unit square S is defined by the IFS

T1(x, y) = (x/2, y/2)

T2(x, y) = (x/2, y/2) + (1/2, 0)

T3(x, y) = (x/2, y/2) + (0, 1/2)

T4(x, y) = (x/2, y/2) + (1/2, 1/2)

That is, S = T₁(S) U T₂(S) U T₃(S) U T₄(S)

Decompose S into regions of address length 1:

S1 = T1(S)

S2 = T2(S)

S3 = T3(S)

S4 = T4(S)

Each of these can be subdivided further, S_ij = T_i(T_j(S)), S_ijk = T_i(T_j(T_k(S))), and so on. In general, the subscript i_n ... i₁ of S_{in ... i1} is the address of this region; note the order of the subscripts is the order of the composition of the T_i defining that region. Here are the regions of address length 2.