Given a continuous map $\gamma$ from $[0,1]$ onto a bounded contractible subset $S$ of $\mathbb R^2$ such that $S$ contains an open subset of $\mathbb R^2$ which is dense in $S$, the preimage $\gamma^{-1}(\partial S)$ of the boundary $\partial S$ should morally be a Cantor set in $[0,1]$ (this should be true for 'natural' examples (Peano curve and its variation by Hilbert, Heighway dragon etc.) but it is easy to make up somewhat artificial examples where $\gamma^{-1}(\partial S)$ contains isolated points and open intervals).
Is there a reference where this problem has been treated?
