Substitution minimal sets

Abstract

Introduction.We show that every substitution on finitely many symbols, which replaces each symbol by a block of length 2 or more, effectively determines at least one almost periodic point under the shift transformation of symbolic dynamics.The orbit-closures of these almost periodic points, called substitution minimal sets, are analyzed topologically to some extent and in particular it is proved that under certain conditions their structure groups are n-adic groups.Several of the known symbolic minimal sets, such as the Morse minimal set, are definable by the present method of construction.Minimal sets exhibiting new properties also appear.The first two sections contain general theorems on the trace relation of transformation groups and on i^-adic transformation groups, some of these results being used to study substitution minimal sets in the third section.A few particular examples of substitution minimal sets are described in the last section.As general references for notions, notation, and terminology occuring here, consult [3; 1].