By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1$ is the projection on the first coordinate. I am curious about the topological entropy of a typical potential space filling curve $f:[0,1]\to [0,1]$.

Are there both examples of vanishing and non vanishing of such potential filling curve?