Substitutions over infinite alphabets and unique ergodicity

Abstract

In extending the study of symbolic substitutions from finite to infinite alphabets, one encounters several obstacles to generalising most of the standard theory. So instead of considering substitutions on arbitrary alphabets, we choose to retain some extra structure from the finite case by demanding that the alphabet carries a compact Hausdorff topology for which the substitution is continuous. Several notions from the classical case can be naturally extended to this setting, such as of a substitution being primitive. Surprisingly, primitivity is no longer sufficient to ensure unique ergodicity of the associated shift space. But we may find conditions which imply unique ergodicity, and which may be verified on a wide range of examples. In place of Perron-Frobenius theory from the finite case, we make use of the theory of positive operators on Banach spaces. This is joint with work Neil Mañibo and Dan Rust.