Bellissard's gap-labelling conjecture

Let ${\Omega}$ be a Cantor set equipped with an action of ${\mathbb Z^p}$ and an invariant probability measure $\mu$. The measure $\mu$ induces a trace $\tau^\mu$ on the crossed-product $C^*$-algebra $C(\Omega)\rtimes\mathbb Z^p$. Let ${\mathbb Z[\mu]}$ be the additive subgroup of $\mathbb R$ generated by the $\mu$-measures of compact-open subsets of ${\Omega}$. Assume that ${\Omega}$ has no non-trivial compact-open invariant subsets.

Bellissard's gap-labelling conjecture.

$$\tau^\mu_*(K_0(C(\Omega)\rtimes\mathbb Z^p))={\mathbb Z[\mu]}.$$

This conjecture predicts the range of the trace on the $K_0$-group of the crossed product and hence the possible integrated density-of-states values in spectral gaps of quasicrystal Hamiltonians. It applies in particular to minimal actions, although the statement here assumes only the absence of non-trivial compact-open invariant subsets.