I was wondering whether most embeddings of one-dimensional subshifts have zero Hausdorff dimension? Given a finite alphabet $\mathcal{A}= \{ 0,...,d-1 \}$ and $\Omega\subseteq \mathcal{A}^\mathbb{N}$, we can encode $\Omega$ to $[0,1]$ via $\omega \overset{\iota}{ \mapsto} \sum_{n=1}^\infty \omega_n \cdot d^{-n}$. I am interseted in the Haudorff dimension of $\iota(\Omega)$, when $\Omega$ is a closed and shift invariant subset.
A result by Furstenberg, which was later generalized by Stephen Simpson in Symbolic dynamics: entropy = dimension = complexity, seems to imply that the Hausdorff dimension is the same as the entropy of a subshift and the Box counting dimension. Given the standard metric on $\mathcal{A}^\mathbb{N}$, it looks like one can generate plenty of $0$-dimensional sets arising naturally. I was under the assumptions that it is hard to come up with natural non-trivial examples of $0$-dimensional sets.
I was wondering whether there is a fault in this reasoning? Or perhaps this should not be surprising at all?