Extending the theory of symbolic substitutions to compact alphabets

Abstract

In this work, joint with Neil Mañibo and Dan Rust, we consider an extension of the theory of symbolic substitutions to infinite alphabets, by requiring the alphabet to carry a compact, Hausdorff topology for which the substitution is continuous. Such substitutions have been considered before, in particular by Durand, Ormes and Petite for zero-dimensional alphabets, and Queffélec in the constant length case. We find a simple condition which ensures that an associated substitution operator is quasi-compact, which we conjecture to always be satisfied for primitive substitutions on countable alphabets. In the primitive case this implies the existence of a unique natural tile length function and, for a recognisable substitution, that the associated shift space is uniquely ergodic. The main tools come from the theory of positive operators on Banach spaces. Very few prerequisites will be assumed, and the theory will be demonstrated via examples.