Bounded-displacement non-equivalence in substitution tilings

Abstract: Given two Delone sets $Y$ and $Z$ in $R^d$ we study the existence of a bounded-displacement (BD) map between them, namely a bijection $f$ from $Y$ to $Z$ so that the quantity $\|y-f(y)\|$, $y\in Y$, is bounded. This notion induces an equivalence relation on collections $X$ of Delone sets and we study the cardinality of BD($X$), a collection of all BD-class representatives. In this talk we focus on sets $X$ of point sets that correspond to tilings in a substitution tiling space. We provide a sufficient condition under which |BD($X$)| is the continuum. In particular we show that, in the context of primitive substitution tilings, |BD($X$)| can be greater than $1$.

dynamical systemsprobability

Audience: researchers in the topic

( slides | video )

Horowitz seminar on probability, ergodic theory and dynamical systems