A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square. 


I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous map $\phi$ from the unit disk $|z|\le 1$ to $\mathbb C^1$ such that 
$\phi$ is holomorphic for $|z|<1$ and the image of the boundary $|z|=1$ has non-empty interior in $\mathbb C^1$ under the map $\phi$.



A "holomorphic" Peano curve?
Here it is:
MR0015154
Salem, R.; Zygmund, A.
Lacunary power series and Peano curves. 
Duke Math. J. 12, (1945). 569–578. 


