Is $\mathrm{Cantor \, set} \times [0,1]$ self-similar?

Question

Consider the Cantor set $\times$ the interval $[0,1]$, i.e. Cantor sets put "one next to the other" as to "cover" the quare $[0,1]\times [0,1]$ is this set self-similar, i.e. the attractor of some homotheties?

Answer 1

Maybe $6$ maps ... map the square $[0,1]\times[0,1]$ onto each of these: $$ \left[0,\frac{1}{3}\right] \times \left[0,\frac{1}{3}\right] \\ \left[\frac{2}{3},1\right] \times \left[0,\frac{1}{3}\right] \\ \left[0,\frac{1}{3}\right] \times \left[\frac{1}{3},\frac{2}{3}\right] \\ \left[\frac{2}{3},1\right] \times \left[\frac{1}{3},\frac{2}{3}\right] \\ \left[0,\frac{1}{3}\right] \times \left[\frac{2}{3},1\right] \\ \left[\frac{2}{3},1\right] \times \left[\frac{2}{3},1\right] $$