Substitutions on compact alphabets

Abstract

To extend symbolic substitutions to infinite alphabets, it is necessary to impose restrictions so as to ensure that important dynamical properties still hold. In place of finiteness of the alphabet we consider alphabets which may be equipped with a compact Hausdorff topology making the substitution a continuous map. For finite alphabets, using Perron–Frobenius Theory, it is well known that primitive substitutions always have a geometric realisation as a substitution of closed, finite intervals of strictly positive lengths and that the associated tiling dynamical system is uniquely ergodic. We extend the first result by proving that primitive substitutions on compact alphabets admit a unique (up to scalar multiplication) continuous natural tile length function. The proof involves consideration of an operator associated to the substitution, which corresponds to the transpose of the standard population matrix from the finite setting. By consideration of the dual operator, we find a class of uniquely ergodic infinite tiling substitution systems, although the question of whether unique ergodicity holds in general for compact alphabets remains open. This is joint work with Neil Mañibo and Dan Rust.