Substitutions over compact alphabets

Abstract

I will introduce the theory of substitutions over compact alphabets, based on joint work with Neil Mañibo and Dan Rust. Combinatorial properties from the finite alphabet case, such as irreducibility and primitivity, can be naturally extended to the topological setting. Similarly, the standard substitution matrix can be replaced, with a substitution operator on a Banach lattice. Existence and uniqueness of natural length functions and ergodic measures on the shift space, which are answered by Perron–Frobenius Theory in the finite setting, become far more delicate. I will explain some recent progress in determining when substitutions over compact alphabets can be guaranteed to produce uniquely ergodic hulls.