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 $g:[0,1]\to [0,1]\times[0,1]$ 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 potetial filling curve?