A General Framework for Tilings, Delone Sets, Functions, and Measures and Their Interrelation

We define a general framework that includes objects such as tilings, Delone sets, functions, and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object $P$ to one that is MLD with $P$ . Moreover, we show that, for an object $P$ and a class $\Sigma$ of objects, a mild condition on them ensures that there exists some $Q\in \Sigma$ that is MLD with $P$ . As an application, we study pattern-equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information on the original object up to MLD, in a quite general setting.