Geodesic
Semi-Groups of Maps in a Locally Compact Space
Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 688-696

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Suppose that S is a locally compact Hausdorff space. A one-parameter semi-group of maps in S is a family {φ t; t ⩾ 0} of continuous functions from S into S satisfying (i) φt0 φu = φt+u for t, u ⩾ 0, where the circle denotes composition, and (ii) φ0 = e, the identity map on S. A semi-group {φt} of maps in S is said to be (iii) of class (C0) if φt(x) → x as t → 0 for each x in S, (iv) separately continuous if the function t → φt(x) is continuous on [0, ∞) for each x in S, and (v) doubly continuous if the function (t, x) → (φt(x) is continuous on [0, ∞) x S. 
Dorroh, J. R. Semi-Groups of Maps in a Locally Compact Space. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 688-696. doi: 10.4153/CJM-1967-063-3
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