Restrictions of harmonic functions and Dirichlet eigenfunctions of the Hata set to the interval

In this paper we study the harmonic functions and the Dirichlet eigenfunctions of the Hata set, and their restrictions to the interval $[0,1]$, its main edge. We prove that these restrictions of the harmonic functions are singular, \ie monotone and with zero derivatives almost everywhere, and provide numerical evidence that this is also the case for the eigenfunctions.