Attractor of Cantor Type with Positive Measure

We construct an iterated function system consisting of strictly increasing contractions $f,g\colon [0,1]\to [0,1]$ with $f([0,1])\cap g([0,1])=\emptyset$ and such that its attractor has positive Lebesgue measure

Around a problem of Nicole Brillouët-Belluot, II

For every α ∈ R we determine all increasing bijections f : (0,+∞) → (0,+∞) such that f(1) = 1 and f(x)f−1(x) = xα for every x ∈ (0,+∞)