Statistical stability for multi-substitution tiling spaces (Q379597)

A \textit{tiling} of \(\mathbb R^d\) is a non-overlapping covering of \(\mathbb R^d\) by tiles. A \textit{multi-substitution tiling} is a tiling determined by a finite number of proto-tiles and a finite set \(\{S_1,\dots,S_k\}\) of substitution maps expressing each proto-tile as a scaling-down of a union of other proto-tiles. Then a choice of \(\overline a\in\{1,\dots,k\}^{\mathbb N}\) determines a \textit{tiling space} \(X_{\overline a}\) as follows: start with a proto-tile; apply \(S_{a_1}\) and inflate, obtaining a finite union of translates of proto-tiles (``patch''); repeat with \(S_{a_2}\), \(S_{a_3}\), etc; and define \(X_{\overline a}\) as the set of tilings of \(\mathbb R^d\) that are limits of translates of these patches.NEWLINENEWLINEThe space \(X_{\overline a}\) is, under mild conditions, a compact uniquely ergodic \(\mathbb R^d\)-space. The unique ergodic measure is closely related to the frequencies of patches: more precisely, the topology of \(X_{\overline a}\) has as basic open neighbourhoods the sets \(X_{p,U}\) (for \(p\) a patch and \(U\) an open in \(\mathbb R^d\)) that consist of tilings containing the patch \(p+u\) for some vector \(u\in U\). The measure of \(X_{p,U}\) is essentially \((\text{the relative frequency of }p)\times\text{vol}(U)\).NEWLINENEWLINEThe main result of the paper (Theorem~5.8) is that the frequencies of patches vary continuously with \(\overline a\in\{1,\dots,k\}^{\mathbb N}\). In some sense, then, the \(\mathbb R^d\)-dynamical system \(X_{\overline a}\) depends continuously on \(\overline a\).