Point-Transitive Actions by the Unit Interval

Abstract

An action is a continuous function α: T × X → X, where T is a semigroup, X is a Hausdorff space, and α(t1, α(t2, x)) = α(t1,t2x) for all t1, t2 ∈ T and x ∈ X . If, for an action α, Q(α) = {x ∈ X| α(T × {x}) = X} is non-empty, then α is called a point-transitive action. Our aim in this note is to classify the point-transitive actions of the unit interval with the usual, nil, or min multiplications.The reader is referred to [5; 7; 9] for information concerning the general theory of semigroups. All semigroups which are considered here are compact and Abelian and all spaces are compact Hausdorff. Actions by semigroups have been studied in [1; 3; 8].