The Structure of Substitution Minimal Sets

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Abstract
Substitutions of constant length on two symbols and their corresponding minimal dynamical systems are divided into three classes: finite, discrete and continuous. Finite substitutions give rise to uninteresting systems. Discrete substitutions yield strictly ergodic systems with discrete spectra, whose topological structure is determined precisely. Continuous substitutions yield strictly ergodic systems with partly continuous and partly discrete spectra, whose topological structure is studied by means of an associated discrete substitution. Topological and measure-theoretic isomorphisms are studied for discrete and continuous substitutions, and a complete topological invariant, the normal form of a substitution, is given.

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