On a one-dimensional analogue of the Smale horseshoe

We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have ${\displaystyle \int \phi \left(T^{n}x\right)f\left(x\right)dx\to \int \phi d\mu }$, where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then ${\displaystyle n^{-1}{\sum }_{i=0}^{n-1}\phi \left(T^{i}x\right)\to \int \phi d\mu }$ for Lebesgue-a.e. x.