Substitutions on infinite compact alphabets

Abstract

I will present recent joint work with Neil Mañibo and Dan Rust on extending the theory of symbolic substitutions to infinite alphabets. To retain some of the flavour of the finite setting, we choose to work with continuous substitutions on alphabets equipped with a compact Hausdorff topology. The most fundamental questions include whether the substitution admits a continuous natural length function and if the resulting two-sided shift space is uniquely ergodic. Unlike in the finite alphabet setting, compact substitutions need not admit a non-zero continuous natural length function, although whether they must for primitive substitutions remains open. It is known that unique ergodicity does not follow from primitivity, by work of Durand, Ormes and Petite. We find a type of coincidence criterion which implies unique ergodicity and seems to apply very generally to primitive substitutions on alphabets containing isolated points. Many results rely on the theory of positive operators on Banach spaces, where the traditional substitution matrix is replaced with the substitution operator on continuous functions over the alphabet.