Geodesic
Point-Transitive Actions by the Unit Interval
Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 255-259

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An action is a continuous function α: T × X → X, where T is a semigroup, X is a Hausdorff space, and α(t 1, α(t 2, x)) = α(t 1,t 2x) for all t 1, t 2 ∈ 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]. 
Borrego, J. T.; DeVun, E. E. Point-Transitive Actions by the Unit Interval. Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 255-259. doi: 10.4153/CJM-1970-033-0
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