Ergodic scales in fractal measures

We will consider a family of fractal measures on the real line $\mathbb{R}$ which are fixed, in the sense of Hutchinson, under a finite family of contractive affine mappings. The maps are chosen such as to leave gaps on $\mathbb{R}$. Hence they have fractal dimension strictly less than $1$. The middle-third Cantor construction is one example. Depending on the gaps and the scaling factor, it is known that the corresponding Hilbert space $L^{2}(\mu)$ exhibits strikingly different properties. In this paper we show that when $\mu$ is fixed in a certain class, there are positive integers $p$ such that multiplication by $p$ modulo $1$ induces an ergodic automorphism on the measure space (support($\mu$), $\mu$).