8 problems

Bellissard's gap-labelling conjecture.

Let $\Omega$ be the Hull of a tiling with a transverse structure for the $\mathbb{R}^d$ action, and suppose the Hull is completely disconnected transversally. Let $\mu$ be an invar…

Gap-labelling equality conjecture. The equality

Gap-opening and gap-labelling conjecture. All possible gaps of $\sigma_\lambda$ are open, and their labels are dyadic numbers or dyadic numbers divided by $3$ in $(0,1)$.

Let $\mathcal T$ be a tiling with finite local complexity. For each patch class $\mathcal P$ of $\mathcal T$, let $\nu(\mathcal P)$ denote its frequency, and let…

Let $\Sigma$ be a Cantor set with a minimal action of $\mathbb{Z}^p$ that preserves a Borel probability measure $\mu$. Let $\sigma$ be the multiplier on $\mathbb{Z}^p$ associated t…

Magnetic gap-labelling conjecture for strongly minimal actions. If the action is strongly minimal, then the containment in the magnetic gap-labelling conjecture for minimal actions…

Magnetic gap-labelling conjecture. The magnetic gap-labelling group is contained in the subgroup of $\mathbb R$ given by